Theorem unen 3338

Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.

Assertion

Ref	Expression
unen	⊢ (((A ≈ B ∧ C ≈ D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D))

Proof of Theorem unen

Step	Hyp	Ref	Expression
1		unexb 1950	. . . . 5 ⊢ ((B ∈ V ∧ D ∈ V) ↔ (B ∪ D) ∈ V)
2		breng 3280	. . . . . 6 ⊢ (B ∈ V → (A ≈ B ↔ ∃f f:A–1-1-onto→B))
3		breng 3280	. . . . . 6 ⊢ (D ∈ V → (C ≈ D ↔ ∃g g:C–1-1-onto→D))
4	2, 3	bi2anan9 478	. . . . 5 ⊢ ((B ∈ V ∧ D ∈ V) → ((A ≈ B ∧ C ≈ D) ↔ (∃f f:A–1-1-onto→B ∧ ∃g g:C–1-1-onto→D)))
5	1, 4	sylbir 176	. . . 4 ⊢ ((B ∪ D) ∈ V → ((A ≈ B ∧ C ≈ D) ↔ (∃f f:A–1-1-onto→B ∧ ∃g g:C–1-1-onto→D)))
6		breng 3280	. . . . . . . 8 ⊢ ((B ∪ D) ∈ V → ((A ∪ C) ≈ (B ∪ D) ↔ ∃h h:(A ∪ C)–1-1-onto→(B ∪ D)))
7		f1oun 2815	. . . . . . . . 9 ⊢ (((f:A–1-1-onto→B ∧ g:C–1-1-onto→D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → (f ∪ g):(A ∪ C)–1-1-onto→(B ∪ D))
8		visset 1350	. . . . . . . . . . 11 ⊢ f ∈ V
9		visset 1350	. . . . . . . . . . 11 ⊢ g ∈ V
10	8, 9	unex 1949	. . . . . . . . . 10 ⊢ (f ∪ g) ∈ V
11		f1oeq1 2795	. . . . . . . . . 10 ⊢ (h = (f ∪ g) → (h:(A ∪ C)–1-1-onto→(B ∪ D) ↔ (f ∪ g):(A ∪ C)–1-1-onto→(B ∪ D)))
12	10, 11	cla4ev 1401	. . . . . . . . 9 ⊢ ((f ∪ g):(A ∪ C)–1-1-onto→(B ∪ D) → ∃h h:(A ∪ C)–1-1-onto→(B ∪ D))
13	7, 12	syl 12	. . . . . . . 8 ⊢ (((f:A–1-1-onto→B ∧ g:C–1-1-onto→D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → ∃h h:(A ∪ C)–1-1-onto→(B ∪ D))
14	6, 13	syl5bir 184	. . . . . . 7 ⊢ ((B ∪ D) ∈ V → (((f:A–1-1-onto→B ∧ g:C–1-1-onto→D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D)))
15	14	exp3a 292	. . . . . 6 ⊢ ((B ∪ D) ∈ V → ((f:A–1-1-onto→B ∧ g:C–1-1-onto→D) → (((A ∩ C) = ∅ ∧ (B ∩ D) = ∅) → (A ∪ C) ≈ (B ∪ D))))
16	15	19.23advv 955	. . . . 5 ⊢ ((B ∪ D) ∈ V → (∃f∃g(f:A–1-1-onto→B ∧ g:C–1-1-onto→D) → (((A ∩ C) = ∅ ∧ (B ∩ D) = ∅) → (A ∪ C) ≈ (B ∪ D))))
17		eeanv 980	. . . . 5 ⊢ (∃f∃g(f:A–1-1-onto→B ∧ g:C–1-1-onto→D) ↔ (∃f f:A–1-1-onto→B ∧ ∃g g:C–1-1-onto→D))
18	16, 17	syl5ibr 182	. . . 4 ⊢ ((B ∪ D) ∈ V → ((∃f f:A–1-1-onto→B ∧ ∃g g:C–1-1-onto→D) → (((A ∩ C) = ∅ ∧ (B ∩ D) = ∅) → (A ∪ C) ≈ (B ∪ D))))
19	5, 18	sylbid 178	. . 3 ⊢ ((B ∪ D) ∈ V → ((A ≈ B ∧ C ≈ D) → (((A ∩ C) = ∅ ∧ (B ∩ D) = ∅) → (A ∪ C) ≈ (B ∪ D))))
20	19	imp3a 279	. 2 ⊢ ((B ∪ D) ∈ V → (((A ≈ B ∧ C ≈ D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D)))
21		brprc 2097	. . . 4 ⊢ (¬ (B ∪ D) ∈ V → ((A ∪ C) ≈ (B ∪ D) ↔ (A ∪ C) ≈ (A ∪ C)))
22		relen 3277	. . . . . . . 8 ⊢ Rel ≈
23	22	brrelexi 2447	. . . . . . 7 ⊢ (A ≈ B → A ∈ V)
24	22	brrelexi 2447	. . . . . . 7 ⊢ (C ≈ D → C ∈ V)
25	23, 24	anim12i 268	. . . . . 6 ⊢ ((A ≈ B ∧ C ≈ D) → (A ∈ V ∧ C ∈ V))
26		unexb 1950	. . . . . 6 ⊢ ((A ∈ V ∧ C ∈ V) ↔ (A ∪ C) ∈ V)
27	25, 26	sylib 173	. . . . 5 ⊢ ((A ≈ B ∧ C ≈ D) → (A ∪ C) ∈ V)
28		enrefg 3294	. . . . 5 ⊢ ((A ∪ C) ∈ V → (A ∪ C) ≈ (A ∪ C))
29	27, 28	syl 12	. . . 4 ⊢ ((A ≈ B ∧ C ≈ D) → (A ∪ C) ≈ (A ∪ C))
30	21, 29	syl5bir 184	. . 3 ⊢ (¬ (B ∪ D) ∈ V → ((A ≈ B ∧ C ≈ D) → (A ∪ C) ≈ (B ∪ D)))
31	30	adantrd 308	. 2 ⊢ (¬ (B ∪ D) ∈ V → (((A ≈ B ∧ C ≈ D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D)))
32	20, 31	pm2.61i 110	1 ⊢ (((A ≈ B ∧ C ≈ D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ∩ cin 1486  ∅c0 1707   class class class wbr 2054  –1-1-onto→wf1o 2421   ≈ cen 3271

This theorem is referenced by:  undom 3342  limensuci 3401  phplem3 3405  pssnn 3428  unfi 3441  infensuc 3484  cdaen 3719  cda1en 3721  cdacomen 3724  cdaassen 3725  xpcdaen 3726

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

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