Theorem unen 6049

Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by set.mm contributors, 11-Jun-1998.)

Assertion

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

Proof of Theorem unen

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oun 5305	. . . . . 6 ⊢ (((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))
2		vex 2863	. . . . . . . 8 ⊢ f ∈ V
3		vex 2863	. . . . . . . 8 ⊢ g ∈ V
4	2, 3	unex 4107	. . . . . . 7 ⊢ (f ∪ g) ∈ V
5		f1oeq1 5282	. . . . . . 7 ⊢ (h = (f ∪ g) → (h:(A ∪ C)–1-1-onto→(B ∪ D) ↔ (f ∪ g):(A ∪ C)–1-1-onto→(B ∪ D)))
6	4, 5	spcev 2947	. . . . . 6 ⊢ ((f ∪ g):(A ∪ C)–1-1-onto→(B ∪ D) → ∃h h:(A ∪ C)–1-1-onto→(B ∪ D))
7	1, 6	syl 15	. . . . 5 ⊢ (((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))
8	7	ex 423	. . . 4 ⊢ ((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)))
9	8	exlimivv 1635	. . 3 ⊢ (∃f∃g(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)))
10	9	imp 418	. 2 ⊢ ((∃f∃g(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))
11		bren 6031	. . . . 5 ⊢ (A ≈ B ↔ ∃f f:A–1-1-onto→B)
12		bren 6031	. . . . 5 ⊢ (C ≈ D ↔ ∃g g:C–1-1-onto→D)
13	11, 12	anbi12i 678	. . . 4 ⊢ ((A ≈ B ∧ C ≈ D) ↔ (∃f f:A–1-1-onto→B ∧ ∃g g:C–1-1-onto→D))
14		eeanv 1913	. . . 4 ⊢ (∃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))
15	13, 14	bitr4i 243	. . 3 ⊢ ((A ≈ B ∧ C ≈ D) ↔ ∃f∃g(f:A–1-1-onto→B ∧ g:C–1-1-onto→D))
16	15	anbi1i 676	. 2 ⊢ (((A ≈ B ∧ C ≈ D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) ↔ (∃f∃g(f:A–1-1-onto→B ∧ g:C–1-1-onto→D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)))
17		bren 6031	. 2 ⊢ ((A ∪ C) ≈ (B ∪ D) ↔ ∃h h:(A ∪ C)–1-1-onto→(B ∪ D))
18	10, 16, 17	3imtr4i 257	1 ⊢ (((A ≈ B ∧ C ≈ D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D))

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∪ cun 3208   ∩ cin 3209  ∅c0 3551   class class class wbr 4640  –1-1-onto→wf1o 4781   ≈ cen 6029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-swap 4725  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-en 6030

This theorem is referenced by:  enadj  6061  ncdisjun  6137  sbthlem1  6204