Theorem undom 3342

Description: Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257.

Hypotheses

Ref	Expression
undom.1	⊢ B ∈ V
undom.2	⊢ C ∈ V
undom.3	⊢ D ∈ V

Assertion

Ref	Expression
undom	⊢ (((A ≼ B ∧ C ≼ D) ∧ (B ∩ D) = ∅) → (A ∪ C) ≼ (B ∪ D))

Proof of Theorem undom

Step	Hyp	Ref	Expression
1		endomtr 3325	. . . . . . . . . . 11 ⊢ (((A ∪ C) ≈ (x ∪ y) ∧ (x ∪ y) ≼ (B ∪ D)) → (A ∪ C) ≼ (B ∪ D))
2		unen 3338	. . . . . . . . . . . . . . 15 ⊢ (((A ≈ x ∧ (C ∖ A) ≈ y) ∧ ((A ∩ (C ∖ A)) = ∅ ∧ (x ∩ y) = ∅)) → (A ∪ (C ∖ A)) ≈ (x ∪ y))
3		undif2 1762	. . . . . . . . . . . . . . . 16 ⊢ (A ∪ (C ∖ A)) = (A ∪ C)
4	3	cleqcomi 1105	. . . . . . . . . . . . . . 15 ⊢ (A ∪ C) = (A ∪ (C ∖ A))
5	2, 4	syl5eqbr 2089	. . . . . . . . . . . . . 14 ⊢ (((A ≈ x ∧ (C ∖ A) ≈ y) ∧ ((A ∩ (C ∖ A)) = ∅ ∧ (x ∩ y) = ∅)) → (A ∪ C) ≈ (x ∪ y))
6		sseq2 1522	. . . . . . . . . . . . . . . . . 18 ⊢ ((B ∩ D) = ∅ → ((x ∩ y) ⊆ (B ∩ D) ↔ (x ∩ y) ⊆ ∅))
7		ss0b 1726	. . . . . . . . . . . . . . . . . 18 ⊢ ((x ∩ y) ⊆ ∅ ↔ (x ∩ y) = ∅)
8	6, 7	syl6bb 414	. . . . . . . . . . . . . . . . 17 ⊢ ((B ∩ D) = ∅ → ((x ∩ y) ⊆ (B ∩ D) ↔ (x ∩ y) = ∅))
9		ss2in 1663	. . . . . . . . . . . . . . . . 17 ⊢ ((x ⊆ B ∧ y ⊆ D) → (x ∩ y) ⊆ (B ∩ D))
10	8, 9	syl5bi 183	. . . . . . . . . . . . . . . 16 ⊢ ((B ∩ D) = ∅ → ((x ⊆ B ∧ y ⊆ D) → (x ∩ y) = ∅))
11	10	imp 277	. . . . . . . . . . . . . . 15 ⊢ (((B ∩ D) = ∅ ∧ (x ⊆ B ∧ y ⊆ D)) → (x ∩ y) = ∅)
12		difdisj 1758	. . . . . . . . . . . . . . 15 ⊢ (A ∩ (C ∖ A)) = ∅
13	11, 12	jctil 240	. . . . . . . . . . . . . 14 ⊢ (((B ∩ D) = ∅ ∧ (x ⊆ B ∧ y ⊆ D)) → ((A ∩ (C ∖ A)) = ∅ ∧ (x ∩ y) = ∅))
14	5, 13	sylan2 346	. . . . . . . . . . . . 13 ⊢ (((A ≈ x ∧ (C ∖ A) ≈ y) ∧ ((B ∩ D) = ∅ ∧ (x ⊆ B ∧ y ⊆ D))) → (A ∪ C) ≈ (x ∪ y))
15	14	anassrs 338	. . . . . . . . . . . 12 ⊢ ((((A ≈ x ∧ (C ∖ A) ≈ y) ∧ (B ∩ D) = ∅) ∧ (x ⊆ B ∧ y ⊆ D)) → (A ∪ C) ≈ (x ∪ y))
16	15	an1rs 373	. . . . . . . . . . 11 ⊢ ((((A ≈ x ∧ (C ∖ A) ≈ y) ∧ (x ⊆ B ∧ y ⊆ D)) ∧ (B ∩ D) = ∅) → (A ∪ C) ≈ (x ∪ y))
17		unss12 1630	. . . . . . . . . . . . 13 ⊢ ((x ⊆ B ∧ y ⊆ D) → (x ∪ y) ⊆ (B ∪ D))
18		undom.1	. . . . . . . . . . . . . . 15 ⊢ B ∈ V
19		undom.3	. . . . . . . . . . . . . . 15 ⊢ D ∈ V
20	18, 19	unex 1949	. . . . . . . . . . . . . 14 ⊢ (B ∪ D) ∈ V
21		ssdom2g 3312	. . . . . . . . . . . . . 14 ⊢ ((B ∪ D) ∈ V → ((x ∪ y) ⊆ (B ∪ D) → (x ∪ y) ≼ (B ∪ D)))
22	20, 21	ax-mp 6	. . . . . . . . . . . . 13 ⊢ ((x ∪ y) ⊆ (B ∪ D) → (x ∪ y) ≼ (B ∪ D))
23	17, 22	syl 12	. . . . . . . . . . . 12 ⊢ ((x ⊆ B ∧ y ⊆ D) → (x ∪ y) ≼ (B ∪ D))
24	23	ad2antlr 321	. . . . . . . . . . 11 ⊢ ((((A ≈ x ∧ (C ∖ A) ≈ y) ∧ (x ⊆ B ∧ y ⊆ D)) ∧ (B ∩ D) = ∅) → (x ∪ y) ≼ (B ∪ D))
25	1, 16, 24	sylanc 361	. . . . . . . . . 10 ⊢ ((((A ≈ x ∧ (C ∖ A) ≈ y) ∧ (x ⊆ B ∧ y ⊆ D)) ∧ (B ∩ D) = ∅) → (A ∪ C) ≼ (B ∪ D))
26	25	exp 291	. . . . . . . . 9 ⊢ (((A ≈ x ∧ (C ∖ A) ≈ y) ∧ (x ⊆ B ∧ y ⊆ D)) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
27	26	an4s 390	. . . . . . . 8 ⊢ (((A ≈ x ∧ x ⊆ B) ∧ ((C ∖ A) ≈ y ∧ y ⊆ D)) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
28	27	exp 291	. . . . . . 7 ⊢ ((A ≈ x ∧ x ⊆ B) → (((C ∖ A) ≈ y ∧ y ⊆ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D))))
29	28	19.23aiv 952	. . . . . 6 ⊢ (∃x(A ≈ x ∧ x ⊆ B) → (((C ∖ A) ≈ y ∧ y ⊆ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D))))
30	29	19.23adv 954	. . . . 5 ⊢ (∃x(A ≈ x ∧ x ⊆ B) → (∃y((C ∖ A) ≈ y ∧ y ⊆ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D))))
31	30	imp 277	. . . 4 ⊢ ((∃x(A ≈ x ∧ x ⊆ B) ∧ ∃y((C ∖ A) ≈ y ∧ y ⊆ D)) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
32	18	domen 3284	. . . 4 ⊢ (A ≼ B ↔ ∃x(A ≈ x ∧ x ⊆ B))
33	19	domen 3284	. . . 4 ⊢ ((C ∖ A) ≼ D ↔ ∃y((C ∖ A) ≈ y ∧ y ⊆ D))
34	31, 32, 33	syl2anb 350	. . 3 ⊢ ((A ≼ B ∧ (C ∖ A) ≼ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
35		undom.2	. . . . 5 ⊢ C ∈ V
36		difss 1596	. . . . 5 ⊢ (C ∖ A) ⊆ C
37		ssdom2g 3312	. . . . 5 ⊢ (C ∈ V → ((C ∖ A) ⊆ C → (C ∖ A) ≼ C))
38	35, 36, 37	mp2 43	. . . 4 ⊢ (C ∖ A) ≼ C
39		domtr 3320	. . . 4 ⊢ (((C ∖ A) ≼ C ∧ C ≼ D) → (C ∖ A) ≼ D)
40	38, 39	mpan 518	. . 3 ⊢ (C ≼ D → (C ∖ A) ≼ D)
41	34, 40	sylan2 346	. 2 ⊢ ((A ≼ B ∧ C ≼ D) → ((B ∩ D) = ∅ → (A ∪ C) ≼ (B ∪ D)))
42	41	imp 277	1 ⊢ (((A ≼ B ∧ C ≼ D) ∧ (B ∩ D) = ∅) → (A ∪ C) ≼ (B ∪ D))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   ≈ cen 3271   ≼ cdom 3272

This theorem is referenced by:  unxpdom2 3651  sucxpdom 3652  uncdadom 3718  cdadom1 3727

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  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-en 3274  df-dom 3275

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