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Theorem unen 6048
 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 (((AB CD) ((AC) = (BD) = )) → (AC) ≈ (BD))

Proof of Theorem unen
Dummy variables f g h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oun 5304 . . . . . 6 (((f:A1-1-ontoB g:C1-1-ontoD) ((AC) = (BD) = )) → (fg):(AC)–1-1-onto→(BD))
2 vex 2862 . . . . . . . 8 f V
3 vex 2862 . . . . . . . 8 g V
42, 3unex 4106 . . . . . . 7 (fg) V
5 f1oeq1 5281 . . . . . . 7 (h = (fg) → (h:(AC)–1-1-onto→(BD) ↔ (fg):(AC)–1-1-onto→(BD)))
64, 5spcev 2946 . . . . . 6 ((fg):(AC)–1-1-onto→(BD) → h h:(AC)–1-1-onto→(BD))
71, 6syl 15 . . . . 5 (((f:A1-1-ontoB g:C1-1-ontoD) ((AC) = (BD) = )) → h h:(AC)–1-1-onto→(BD))
87ex 423 . . . 4 ((f:A1-1-ontoB g:C1-1-ontoD) → (((AC) = (BD) = ) → h h:(AC)–1-1-onto→(BD)))
98exlimivv 1635 . . 3 (fg(f:A1-1-ontoB g:C1-1-ontoD) → (((AC) = (BD) = ) → h h:(AC)–1-1-onto→(BD)))
109imp 418 . 2 ((fg(f:A1-1-ontoB g:C1-1-ontoD) ((AC) = (BD) = )) → h h:(AC)–1-1-onto→(BD))
11 bren 6030 . . . . 5 (ABf f:A1-1-ontoB)
12 bren 6030 . . . . 5 (CDg g:C1-1-ontoD)
1311, 12anbi12i 678 . . . 4 ((AB CD) ↔ (f f:A1-1-ontoB g g:C1-1-ontoD))
14 eeanv 1913 . . . 4 (fg(f:A1-1-ontoB g:C1-1-ontoD) ↔ (f f:A1-1-ontoB g g:C1-1-ontoD))
1513, 14bitr4i 243 . . 3 ((AB CD) ↔ fg(f:A1-1-ontoB g:C1-1-ontoD))
1615anbi1i 676 . 2 (((AB CD) ((AC) = (BD) = )) ↔ (fg(f:A1-1-ontoB g:C1-1-ontoD) ((AC) = (BD) = )))
17 bren 6030 . 2 ((AC) ≈ (BD) ↔ h h:(AC)–1-1-onto→(BD))
1810, 16, 173imtr4i 257 1 (((AB CD) ((AC) = (BD) = )) → (AC) ≈ (BD))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∪ cun 3207   ∩ cin 3208  ∅c0 3550   class class class wbr 4639  –1-1-onto→wf1o 4780   ≈ cen 6028 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-swap 4724  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-en 6029 This theorem is referenced by:  enadj  6060  ncdisjun  6136  sbthlem1  6203
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