Theorem atcvat4 5775

Description: A condition implying existence of an atom with the properties shown. Lemma 3.2.20 of [PtakPulmannova] p. 68.

Hypothesis

Ref	Expression
atcvat3.1	⊢ A ∈ Cℋ

Assertion

Ref	Expression
atcvat4	⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((¬ A = 0ℋ ∧ B ⊆ (A ∨ℋ C)) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x))))

Distinct variable group(s):   x,A   x,B   x,C

Proof of Theorem atcvat4

Step	Hyp	Ref	Expression
1		sseq1 1521	. . . . . . . . . . . . . . . 16 ⊢ (B = C → (B ⊆ (C ∨ℋ x) ↔ C ⊆ (C ∨ℋ x)))
2		chub1t 5424	. . . . . . . . . . . . . . . . 17 ⊢ ((C ∈ Cℋ ∧ x ∈ Cℋ ) → C ⊆ (C ∨ℋ x))
3		atelch 5742	. . . . . . . . . . . . . . . . 17 ⊢ (C ∈ Atoms → C ∈ Cℋ )
4		atelch 5742	. . . . . . . . . . . . . . . . 17 ⊢ (x ∈ Atoms → x ∈ Cℋ )
5	2, 3, 4	syl2an 349	. . . . . . . . . . . . . . . 16 ⊢ ((C ∈ Atoms ∧ x ∈ Atoms) → C ⊆ (C ∨ℋ x))
6	1, 5	syl5bir 184	. . . . . . . . . . . . . . 15 ⊢ (B = C → ((C ∈ Atoms ∧ x ∈ Atoms) → B ⊆ (C ∨ℋ x)))
7	6	exp3a 292	. . . . . . . . . . . . . 14 ⊢ (B = C → (C ∈ Atoms → (x ∈ Atoms → B ⊆ (C ∨ℋ x))))
8	7	com12 13	. . . . . . . . . . . . 13 ⊢ (C ∈ Atoms → (B = C → (x ∈ Atoms → B ⊆ (C ∨ℋ x))))
9	8	imp 277	. . . . . . . . . . . 12 ⊢ ((C ∈ Atoms ∧ B = C) → (x ∈ Atoms → B ⊆ (C ∨ℋ x)))
10	9	anim2d 433	. . . . . . . . . . 11 ⊢ ((C ∈ Atoms ∧ B = C) → ((x ⊆ A ∧ x ∈ Atoms) → (x ⊆ A ∧ B ⊆ (C ∨ℋ x))))
11	10	exp3a 292	. . . . . . . . . 10 ⊢ ((C ∈ Atoms ∧ B = C) → (x ⊆ A → (x ∈ Atoms → (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))))
12	11	com23 32	. . . . . . . . 9 ⊢ ((C ∈ Atoms ∧ B = C) → (x ∈ Atoms → (x ⊆ A → (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))))
13	12	r19.22dv 1278	. . . . . . . 8 ⊢ ((C ∈ Atoms ∧ B = C) → (∃x ∈ Atoms x ⊆ A → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x))))
14		atcvat3.1	. . . . . . . . 9 ⊢ A ∈ Cℋ
15	14	hatomic 5754	. . . . . . . 8 ⊢ (¬ A = 0ℋ → ∃x ∈ Atoms x ⊆ A)
16	13, 15	syl5 22	. . . . . . 7 ⊢ ((C ∈ Atoms ∧ B = C) → (¬ A = 0ℋ → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x))))
17	16	exp 291	. . . . . 6 ⊢ (C ∈ Atoms → (B = C → (¬ A = 0ℋ → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))))
18	17	a1i 7	. . . . 5 ⊢ (B ⊆ (A ∨ℋ C) → (C ∈ Atoms → (B = C → (¬ A = 0ℋ → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x))))))
19	18	com4l 39	. . . 4 ⊢ (C ∈ Atoms → (B = C → (¬ A = 0ℋ → (B ⊆ (A ∨ℋ C) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x))))))
20	19	imp4a 282	. . 3 ⊢ (C ∈ Atoms → (B = C → ((¬ A = 0ℋ ∧ B ⊆ (A ∨ℋ C)) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))))
21	20	adantl 305	. 2 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (B = C → ((¬ A = 0ℋ ∧ B ⊆ (A ∨ℋ C)) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))))
22		pm3.26 256	. . . . . . . . 9 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → B ∈ Atoms)
23	22	a1d 14	. . . . . . . 8 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → B ∈ Atoms))
24		chlejb2t 5430	. . . . . . . . . . . . . . . . 17 ⊢ ((C ∈ Cℋ ∧ A ∈ Cℋ ) → (C ⊆ A ↔ (A ∨ℋ C) = A))
25	14, 24	mpan2 519	. . . . . . . . . . . . . . . 16 ⊢ (C ∈ Cℋ → (C ⊆ A ↔ (A ∨ℋ C) = A))
26	25	biimpa 324	. . . . . . . . . . . . . . 15 ⊢ ((C ∈ Cℋ ∧ C ⊆ A) → (A ∨ℋ C) = A)
27	26	sseq2d 1528	. . . . . . . . . . . . . 14 ⊢ ((C ∈ Cℋ ∧ C ⊆ A) → (B ⊆ (A ∨ℋ C) ↔ B ⊆ A))
28	27	biimpa 324	. . . . . . . . . . . . 13 ⊢ (((C ∈ Cℋ ∧ C ⊆ A) ∧ B ⊆ (A ∨ℋ C)) → B ⊆ A)
29	28	anasss 337	. . . . . . . . . . . 12 ⊢ ((C ∈ Cℋ ∧ (C ⊆ A ∧ B ⊆ (A ∨ℋ C))) → B ⊆ A)
30	29	exp 291	. . . . . . . . . . 11 ⊢ (C ∈ Cℋ → ((C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → B ⊆ A))
31	30	adantl 305	. . . . . . . . . 10 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → ((C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → B ⊆ A))
32		chub2t 5425	. . . . . . . . . . 11 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → B ⊆ (C ∨ℋ B))
33	32	a1d 14	. . . . . . . . . 10 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → ((C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → B ⊆ (C ∨ℋ B)))
34	31, 33	jcad 455	. . . . . . . . 9 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → ((C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → (B ⊆ A ∧ B ⊆ (C ∨ℋ B))))
35		atelch 5742	. . . . . . . . 9 ⊢ (B ∈ Atoms → B ∈ Cℋ )
36	34, 35, 3	syl2an 349	. . . . . . . 8 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → (B ⊆ A ∧ B ⊆ (C ∨ℋ B))))
37	23, 36	jcad 455	. . . . . . 7 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → (B ∈ Atoms ∧ (B ⊆ A ∧ B ⊆ (C ∨ℋ B)))))
38	37	imp 277	. . . . . 6 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ (C ⊆ A ∧ B ⊆ (A ∨ℋ C))) → (B ∈ Atoms ∧ (B ⊆ A ∧ B ⊆ (C ∨ℋ B))))
39	38	anassrs 338	. . . . 5 ⊢ ((((B ∈ Atoms ∧ C ∈ Atoms) ∧ C ⊆ A) ∧ B ⊆ (A ∨ℋ C)) → (B ∈ Atoms ∧ (B ⊆ A ∧ B ⊆ (C ∨ℋ B))))
40		sseq1 1521	. . . . . . 7 ⊢ (x = B → (x ⊆ A ↔ B ⊆ A))
41		opreq2 3007	. . . . . . . 8 ⊢ (x = B → (C ∨ℋ x) = (C ∨ℋ B))
42	41	sseq2d 1528	. . . . . . 7 ⊢ (x = B → (B ⊆ (C ∨ℋ x) ↔ B ⊆ (C ∨ℋ B)))
43	40, 42	anbi12d 476	. . . . . 6 ⊢ (x = B → ((x ⊆ A ∧ B ⊆ (C ∨ℋ x)) ↔ (B ⊆ A ∧ B ⊆ (C ∨ℋ B))))
44	43	rcla4ev 1403	. . . . 5 ⊢ ((B ∈ Atoms ∧ (B ⊆ A ∧ B ⊆ (C ∨ℋ B))) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))
45	39, 44	syl 12	. . . 4 ⊢ ((((B ∈ Atoms ∧ C ∈ Atoms) ∧ C ⊆ A) ∧ B ⊆ (A ∨ℋ C)) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))
46	45	adantrl 311	. . 3 ⊢ ((((B ∈ Atoms ∧ C ∈ Atoms) ∧ C ⊆ A) ∧ (¬ A = 0ℋ ∧ B ⊆ (A ∨ℋ C))) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))
47	46	exp31 293	. 2 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (C ⊆ A → ((¬ A = 0ℋ ∧ B ⊆ (A ∨ℋ C)) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))))
48	14	atcvat3 5774	. . . . . . 7 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C)) → (A ∩ (B ∨ℋ C)) ∈ Atoms))
49		atexch 5769	. . . . . . . . . 10 ⊢ ((C ∈ Cℋ ∧ (A ∩ (B ∨ℋ C)) ∈ Atoms ∧ B ∈ Atoms) → (((A ∩ (B ∨ℋ C)) ⊆ (C ∨ℋ B) ∧ (C ∩ (A ∩ (B ∨ℋ C))) = 0ℋ) → B ⊆ (C ∨ℋ (A ∩ (B ∨ℋ C)))))
50	3	ad2antlr 321	. . . . . . . . . . 11 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C))) → C ∈ Cℋ )
51	48	imp 277	. . . . . . . . . . 11 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C))) → (A ∩ (B ∨ℋ C)) ∈ Atoms)
52	22	adantr 306	. . . . . . . . . . 11 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C))) → B ∈ Atoms)
53	50, 51, 52	3jca 604	. . . . . . . . . 10 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C))) → (C ∈ Cℋ ∧ (A ∩ (B ∨ℋ C)) ∈ Atoms ∧ B ∈ Atoms))
54		inss2 1658	. . . . . . . . . . . . . 14 ⊢ (A ∩ (B ∨ℋ C)) ⊆ (B ∨ℋ C)
55	54	a1i 7	. . . . . . . . . . . . 13 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (A ∩ (B ∨ℋ C)) ⊆ (B ∨ℋ C))
56		chjcomt 5423	. . . . . . . . . . . . . 14 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (B ∨ℋ C) = (C ∨ℋ B))
57	56, 35, 3	syl2an 349	. . . . . . . . . . . . 13 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (B ∨ℋ C) = (C ∨ℋ B))
58	55, 57	sseqtrd 1536	. . . . . . . . . . . 12 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (A ∩ (B ∨ℋ C)) ⊆ (C ∨ℋ B))
59	58	adantr 306	. . . . . . . . . . 11 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C))) → (A ∩ (B ∨ℋ C)) ⊆ (C ∨ℋ B))
60		atnssm0 5765	. . . . . . . . . . . . . . . . 17 ⊢ ((A ∈ Cℋ ∧ C ∈ Atoms) → (¬ C ⊆ A ↔ (A ∩ C) = 0ℋ))
61	14, 60	mpan 518	. . . . . . . . . . . . . . . 16 ⊢ (C ∈ Atoms → (¬ C ⊆ A ↔ (A ∩ C) = 0ℋ))
62	61	adantl 305	. . . . . . . . . . . . . . 15 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (¬ C ⊆ A ↔ (A ∩ C) = 0ℋ))
63		chinclt 5416	. . . . . . . . . . . . . . . . . . 19 ⊢ ((C ∈ Cℋ ∧ (A ∩ (B ∨ℋ C)) ∈ Cℋ ) → (C ∩ (A ∩ (B ∨ℋ C))) ∈ Cℋ )
64		pm3.27 260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → C ∈ Cℋ )
65		chjclt 5330	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (B ∨ℋ C) ∈ Cℋ )
66		chinclt 5416	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((A ∈ Cℋ ∧ (B ∨ℋ C) ∈ Cℋ ) → (A ∩ (B ∨ℋ C)) ∈ Cℋ )
67	14, 66	mpan 518	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((B ∨ℋ C) ∈ Cℋ → (A ∩ (B ∨ℋ C)) ∈ Cℋ )
68	65, 67	syl 12	. . . . . . . . . . . . . . . . . . 19 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ∩ (B ∨ℋ C)) ∈ Cℋ )
69	63, 64, 68	sylanc 361	. . . . . . . . . . . . . . . . . 18 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (C ∩ (A ∩ (B ∨ℋ C))) ∈ Cℋ )
70	69, 35, 3	syl2an 349	. . . . . . . . . . . . . . . . 17 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (C ∩ (A ∩ (B ∨ℋ C))) ∈ Cℋ )
71		chle0t 5368	. . . . . . . . . . . . . . . . 17 ⊢ ((C ∩ (A ∩ (B ∨ℋ C))) ∈ Cℋ → ((C ∩ (A ∩ (B ∨ℋ C))) ⊆ 0ℋ ↔ (C ∩ (A ∩ (B ∨ℋ C))) = 0ℋ))
72	70, 71	syl 12	. . . . . . . . . . . . . . . 16 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((C ∩ (A ∩ (B ∨ℋ C))) ⊆ 0ℋ ↔ (C ∩ (A ∩ (B ∨ℋ C))) = 0ℋ))
73		inss1 1657	. . . . . . . . . . . . . . . . . . 19 ⊢ (A ∩ (B ∨ℋ C)) ⊆ A
74		sslin 1662	. . . . . . . . . . . . . . . . . . 19 ⊢ ((A ∩ (B ∨ℋ C)) ⊆ A → (C ∩ (A ∩ (B ∨ℋ C))) ⊆ (C ∩ A))
75	73, 74	ax-mp 6	. . . . . . . . . . . . . . . . . 18 ⊢ (C ∩ (A ∩ (B ∨ℋ C))) ⊆ (C ∩ A)
76		incom 1636	. . . . . . . . . . . . . . . . . 18 ⊢ (C ∩ A) = (A ∩ C)
77	75, 76	sseqtr 1532	. . . . . . . . . . . . . . . . 17 ⊢ (C ∩ (A ∩ (B ∨ℋ C))) ⊆ (A ∩ C)
78		sseq2 1522	. . . . . . . . . . . . . . . . 17 ⊢ ((A ∩ C) = 0ℋ → ((C ∩ (A ∩ (B ∨ℋ C))) ⊆ (A ∩ C) ↔ (C ∩ (A ∩ (B ∨ℋ C))) ⊆ 0ℋ))
79	77, 78	mpbii 168	. . . . . . . . . . . . . . . 16 ⊢ ((A ∩ C) = 0ℋ → (C ∩ (A ∩ (B ∨ℋ C))) ⊆ 0ℋ)
80	72, 79	syl5bi 183	. . . . . . . . . . . . . . 15 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((A ∩ C) = 0ℋ → (C ∩ (A ∩ (B ∨ℋ C))) = 0ℋ))
81	62, 80	sylbid 178	. . . . . . . . . . . . . 14 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (¬ C ⊆ A → (C ∩ (A ∩ (B ∨ℋ C))) = 0ℋ))
82	81	imp 277	. . . . . . . . . . . . 13 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ¬ C ⊆ A) → (C ∩ (A ∩ (B ∨ℋ C))) = 0ℋ)
83	82	adantrl 311	. . . . . . . . . . . 12 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ (¬ B = C ∧ ¬ C ⊆ A)) → (C ∩ (A ∩ (B ∨ℋ C))) = 0ℋ)
84	83	adantrr 312	. . . . . . . . . . 11 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C))) → (C ∩ (A ∩ (B ∨ℋ C))) = 0ℋ)
85	59, 84	jca 236	. . . . . . . . . 10 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C))) → ((A ∩ (B ∨ℋ C)) ⊆ (C ∨ℋ B) ∧ (C ∩ (A ∩ (B ∨ℋ C))) = 0ℋ))
86	49, 53, 85	sylc 62	. . . . . . . . 9 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C))) → B ⊆ (C ∨ℋ (A ∩ (B ∨ℋ C))))
87	86, 73	jctil 240	. . . . . . . 8 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C))) → ((A ∩ (B ∨ℋ C)) ⊆ A ∧ B ⊆ (C ∨ℋ (A ∩ (B ∨ℋ C)))))
88	87	exp 291	. . . . . . 7 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C)) → ((A ∩ (B ∨ℋ C)) ⊆ A ∧ B ⊆ (C ∨ℋ (A ∩ (B ∨ℋ C))))))
89	48, 88	jcad 455	. . . . . 6 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C)) → ((A ∩ (B ∨ℋ C)) ∈ Atoms ∧ ((A ∩ (B ∨ℋ C)) ⊆ A ∧ B ⊆ (C ∨ℋ (A ∩ (B ∨ℋ C)))))))
90		sseq1 1521	. . . . . . . 8 ⊢ (x = (A ∩ (B ∨ℋ C)) → (x ⊆ A ↔ (A ∩ (B ∨ℋ C)) ⊆ A))
91		opreq2 3007	. . . . . . . . 9 ⊢ (x = (A ∩ (B ∨ℋ C)) → (C ∨ℋ x) = (C ∨ℋ (A ∩ (B ∨ℋ C))))
92	91	sseq2d 1528	. . . . . . . 8 ⊢ (x = (A ∩ (B ∨ℋ C)) → (B ⊆ (C ∨ℋ x) ↔ B ⊆ (C ∨ℋ (A ∩ (B ∨ℋ C)))))
93	90, 92	anbi12d 476	. . . . . . 7 ⊢ (x = (A ∩ (B ∨ℋ C)) → ((x ⊆ A ∧ B ⊆ (C ∨ℋ x)) ↔ ((A ∩ (B ∨ℋ C)) ⊆ A ∧ B ⊆ (C ∨ℋ (A ∩ (B ∨ℋ C))))))
94	93	rcla4ev 1403	. . . . . 6 ⊢ (((A ∩ (B ∨ℋ C)) ∈ Atoms ∧ ((A ∩ (B ∨ℋ C)) ⊆ A ∧ B ⊆ (C ∨ℋ (A ∩ (B ∨ℋ C))))) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))
95	89, 94	syl6 23	. . . . 5 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C)) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x))))
96	95	exp3a 292	. . . 4 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((¬ B = C ∧ ¬ C ⊆ A) → (B ⊆ (A ∨ℋ C) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))))
97		ioran 254	. . . 4 ⊢ (¬ (B = C ∨ C ⊆ A) ↔ (¬ B = C ∧ ¬ C ⊆ A))
98	96, 97	syl5ib 181	. . 3 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (¬ (B = C ∨ C ⊆ A) → (B ⊆ (A ∨ℋ C) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))))
99		pm3.27 260	. . 3 ⊢ ((¬ A = 0ℋ ∧ B ⊆ (A ∨ℋ C)) → B ⊆ (A ∨ℋ C))
100	98, 99	syl7 24	. 2 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (¬ (B = C ∨ C ⊆ A) → ((¬ A = 0ℋ ∧ B ⊆ (A ∨ℋ C)) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x)))))
101	21, 47, 100	ecase3d 560	1 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((¬ A = 0ℋ ∧ B ⊆ (A ∨ℋ C)) → ∃x ∈ Atoms (x ⊆ A ∧ B ⊆ (C ∨ℋ x))))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ∩ cin 1486   ⊆ wss 1487  (class class class)co 3001   C_ℋ cch 4968   ∨_ℋ chj 4972  0_ℋc0h 4974  Atomscat 4980

This theorem is referenced by:  mdsymlem3 5778

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080  ax-hilex 4983  ax-hvaddcl 4984  ax-hvcom 4985  ax-hvass 4986  ax-hvzercl 4987  ax-hvaddid 4988  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvdistr2 4994  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  ax-his3 5047  ax-his4 5048  ax-hcompl 5113

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-4 4464  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676  df-sqr 4728  df-re 4790  df-im 4791  df-cj 4792  df-abs 4793  df-clim 4876  df-hvsub 4996  df-hnorm 5074  df-cauchy 5102  df-hlim 5107  df-sh 5114  df-ch 5127  df-oc 5156  df-ch0 5157  df-pj 5244  df-shsum 5275  df-span 5276  df-chj 5277  df-chsup 5278  df-cv 5712  df-at 5737

metamath.org