Theorem unfilem1 3438

Description: Lemma for proving that the union of two finite sets is finite.

Hypotheses

Ref	Expression
unfilem1.1	⊢ A ∈ ω
unfilem1.2	⊢ B ∈ ω
unfilem1.3	⊢ F = {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}

Assertion

Ref	Expression
unfilem1	⊢ ran F = ((A +o B) ∖ A)

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

Proof of Theorem unfilem1

Step	Hyp	Ref	Expression
1		rnopab 2566	. 2 ⊢ ran {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))} = {y∣∃x(x ∈ B ∧ y = (A +o x))}
2		unfilem1.3	. . 3 ⊢ F = {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}
3	2	rneqi 2556	. 2 ⊢ ran F = ran {⟨x, y⟩∣(x ∈ B ∧ y = (A +o x))}
4		eldif 1496	. . . 4 ⊢ (y ∈ ((A +o B) ∖ A) ↔ (y ∈ (A +o B) ∧ ¬ y ∈ A))
5		unfilem1.1	. . . . . . . . . 10 ⊢ A ∈ ω
6		unfilem1.2	. . . . . . . . . 10 ⊢ B ∈ ω
7		nnacl 3172	. . . . . . . . . 10 ⊢ ((A ∈ ω ∧ B ∈ ω) → (A +o B) ∈ ω)
8	5, 6, 7	mp2an 520	. . . . . . . . 9 ⊢ (A +o B) ∈ ω
9		elnn 2383	. . . . . . . . 9 ⊢ ((y ∈ (A +o B) ∧ (A +o B) ∈ ω) → y ∈ ω)
10	8, 9	mpan2 519	. . . . . . . 8 ⊢ (y ∈ (A +o B) → y ∈ ω)
11		ordtri1 2231	. . . . . . . . . . . 12 ⊢ ((Ord A ∧ Ord y) → (A ⊆ y ↔ ¬ y ∈ A))
12		nnord 2381	. . . . . . . . . . . 12 ⊢ (A ∈ ω → Ord A)
13		nnord 2381	. . . . . . . . . . . 12 ⊢ (y ∈ ω → Ord y)
14	11, 12, 13	syl2an 349	. . . . . . . . . . 11 ⊢ ((A ∈ ω ∧ y ∈ ω) → (A ⊆ y ↔ ¬ y ∈ A))
15		nnawordex 3192	. . . . . . . . . . 11 ⊢ ((A ∈ ω ∧ y ∈ ω) → (A ⊆ y ↔ ∃x ∈ ω (A +o x) = y))
16	14, 15	bitr3d 408	. . . . . . . . . 10 ⊢ ((A ∈ ω ∧ y ∈ ω) → (¬ y ∈ A ↔ ∃x ∈ ω (A +o x) = y))
17	5, 16	mpan 518	. . . . . . . . 9 ⊢ (y ∈ ω → (¬ y ∈ A ↔ ∃x ∈ ω (A +o x) = y))
18		df-rex 1206	. . . . . . . . 9 ⊢ (∃x ∈ ω (A +o x) = y ↔ ∃x(x ∈ ω ∧ (A +o x) = y))
19	17, 18	syl6bb 414	. . . . . . . 8 ⊢ (y ∈ ω → (¬ y ∈ A ↔ ∃x(x ∈ ω ∧ (A +o x) = y)))
20	10, 19	syl 12	. . . . . . 7 ⊢ (y ∈ (A +o B) → (¬ y ∈ A ↔ ∃x(x ∈ ω ∧ (A +o x) = y)))
21		nnaord 3177	. . . . . . . . . . . . 13 ⊢ ((x ∈ ω ∧ B ∈ ω ∧ A ∈ ω) → (x ∈ B ↔ (A +o x) ∈ (A +o B)))
22	5, 21	mp3an3 641	. . . . . . . . . . . 12 ⊢ ((x ∈ ω ∧ B ∈ ω) → (x ∈ B ↔ (A +o x) ∈ (A +o B)))
23	6, 22	mpan2 519	. . . . . . . . . . 11 ⊢ (x ∈ ω → (x ∈ B ↔ (A +o x) ∈ (A +o B)))
24		eleq1 1149	. . . . . . . . . . 11 ⊢ ((A +o x) = y → ((A +o x) ∈ (A +o B) ↔ y ∈ (A +o B)))
25	23, 24	sylan9bb 418	. . . . . . . . . 10 ⊢ ((x ∈ ω ∧ (A +o x) = y) → (x ∈ B ↔ y ∈ (A +o B)))
26	25	biimprcd 138	. . . . . . . . 9 ⊢ (y ∈ (A +o B) → ((x ∈ ω ∧ (A +o x) = y) → x ∈ B))
27		cleqcom 1103	. . . . . . . . . . . 12 ⊢ ((A +o x) = y ↔ y = (A +o x))
28	27	biimp 133	. . . . . . . . . . 11 ⊢ ((A +o x) = y → y = (A +o x))
29	28	adantl 305	. . . . . . . . . 10 ⊢ ((x ∈ ω ∧ (A +o x) = y) → y = (A +o x))
30	29	a1i 7	. . . . . . . . 9 ⊢ (y ∈ (A +o B) → ((x ∈ ω ∧ (A +o x) = y) → y = (A +o x)))
31	26, 30	jcad 455	. . . . . . . 8 ⊢ (y ∈ (A +o B) → ((x ∈ ω ∧ (A +o x) = y) → (x ∈ B ∧ y = (A +o x))))
32	31	19.22dv 947	. . . . . . 7 ⊢ (y ∈ (A +o B) → (∃x(x ∈ ω ∧ (A +o x) = y) → ∃x(x ∈ B ∧ y = (A +o x))))
33	20, 32	sylbid 178	. . . . . 6 ⊢ (y ∈ (A +o B) → (¬ y ∈ A → ∃x(x ∈ B ∧ y = (A +o x))))
34	33	imp 277	. . . . 5 ⊢ ((y ∈ (A +o B) ∧ ¬ y ∈ A) → ∃x(x ∈ B ∧ y = (A +o x)))
35		eleq1 1149	. . . . . . . . 9 ⊢ (y = (A +o x) → (y ∈ (A +o B) ↔ (A +o x) ∈ (A +o B)))
36		eleq1 1149	. . . . . . . . . 10 ⊢ (y = (A +o x) → (y ∈ A ↔ (A +o x) ∈ A))
37	36	negbid 463	. . . . . . . . 9 ⊢ (y = (A +o x) → (¬ y ∈ A ↔ ¬ (A +o x) ∈ A))
38	35, 37	anbi12d 476	. . . . . . . 8 ⊢ (y = (A +o x) → ((y ∈ (A +o B) ∧ ¬ y ∈ A) ↔ ((A +o x) ∈ (A +o B) ∧ ¬ (A +o x) ∈ A)))
39	38	biimparc 327	. . . . . . 7 ⊢ ((((A +o x) ∈ (A +o B) ∧ ¬ (A +o x) ∈ A) ∧ y = (A +o x)) → (y ∈ (A +o B) ∧ ¬ y ∈ A))
40		elnn 2383	. . . . . . . . . . 11 ⊢ ((x ∈ B ∧ B ∈ ω) → x ∈ ω)
41	6, 40	mpan2 519	. . . . . . . . . 10 ⊢ (x ∈ B → x ∈ ω)
42	41, 23	syl 12	. . . . . . . . 9 ⊢ (x ∈ B → (x ∈ B ↔ (A +o x) ∈ (A +o B)))
43	42	ibi 449	. . . . . . . 8 ⊢ (x ∈ B → (A +o x) ∈ (A +o B))
44		nnaword1 3186	. . . . . . . . . . 11 ⊢ ((A ∈ ω ∧ x ∈ ω) → A ⊆ (A +o x))
45		nnacl 3172	. . . . . . . . . . . 12 ⊢ ((A ∈ ω ∧ x ∈ ω) → (A +o x) ∈ ω)
46		nnord 2381	. . . . . . . . . . . 12 ⊢ ((A +o x) ∈ ω → Ord (A +o x))
47	5, 12	ax-mp 6	. . . . . . . . . . . . 13 ⊢ Ord A
48		ordtri1 2231	. . . . . . . . . . . . 13 ⊢ ((Ord A ∧ Ord (A +o x)) → (A ⊆ (A +o x) ↔ ¬ (A +o x) ∈ A))
49	47, 48	mpan 518	. . . . . . . . . . . 12 ⊢ (Ord (A +o x) → (A ⊆ (A +o x) ↔ ¬ (A +o x) ∈ A))
50	45, 46, 49	3syl 21	. . . . . . . . . . 11 ⊢ ((A ∈ ω ∧ x ∈ ω) → (A ⊆ (A +o x) ↔ ¬ (A +o x) ∈ A))
51	44, 50	mpbid 170	. . . . . . . . . 10 ⊢ ((A ∈ ω ∧ x ∈ ω) → ¬ (A +o x) ∈ A)
52	5, 51	mpan 518	. . . . . . . . 9 ⊢ (x ∈ ω → ¬ (A +o x) ∈ A)
53	41, 52	syl 12	. . . . . . . 8 ⊢ (x ∈ B → ¬ (A +o x) ∈ A)
54	43, 53	jca 236	. . . . . . 7 ⊢ (x ∈ B → ((A +o x) ∈ (A +o B) ∧ ¬ (A +o x) ∈ A))
55	39, 54	sylan 343	. . . . . 6 ⊢ ((x ∈ B ∧ y = (A +o x)) → (y ∈ (A +o B) ∧ ¬ y ∈ A))
56	55	19.23aiv 952	. . . . 5 ⊢ (∃x(x ∈ B ∧ y = (A +o x)) → (y ∈ (A +o B) ∧ ¬ y ∈ A))
57	34, 56	impbi 139	. . . 4 ⊢ ((y ∈ (A +o B) ∧ ¬ y ∈ A) ↔ ∃x(x ∈ B ∧ y = (A +o x)))
58	4, 57	bitr 151	. . 3 ⊢ (y ∈ ((A +o B) ∖ A) ↔ ∃x(x ∈ B ∧ y = (A +o x)))
59	58	biabri 1180	. 2 ⊢ ((A +o B) ∖ A) = {y∣∃x(x ∈ B ∧ y = (A +o x))}
60	1, 3, 59	3eqtr4 1126	1 ⊢ ran F = ((A +o B) ∖ A)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ∖ cdif 1484   ⊆ wss 1487  {copab 2055  Ord word 2198  ωcom 2372  ran crn 2411  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  unfilem2 3439

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-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-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-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-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-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106

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