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Theorem unen 8583
Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
unen (((𝐴𝐵𝐶𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐴𝐶) ≈ (𝐵𝐷))

Proof of Theorem unen
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bren 8505 . . 3 (𝐴𝐵 ↔ ∃𝑥 𝑥:𝐴1-1-onto𝐵)
2 bren 8505 . . 3 (𝐶𝐷 ↔ ∃𝑦 𝑦:𝐶1-1-onto𝐷)
3 exdistrv 1956 . . . 4 (∃𝑥𝑦(𝑥:𝐴1-1-onto𝐵𝑦:𝐶1-1-onto𝐷) ↔ (∃𝑥 𝑥:𝐴1-1-onto𝐵 ∧ ∃𝑦 𝑦:𝐶1-1-onto𝐷))
4 vex 3447 . . . . . . . 8 𝑥 ∈ V
5 vex 3447 . . . . . . . 8 𝑦 ∈ V
64, 5unex 7453 . . . . . . 7 (𝑥𝑦) ∈ V
7 f1oun 6613 . . . . . . 7 (((𝑥:𝐴1-1-onto𝐵𝑦:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝑥𝑦):(𝐴𝐶)–1-1-onto→(𝐵𝐷))
8 f1oen3g 8512 . . . . . . 7 (((𝑥𝑦) ∈ V ∧ (𝑥𝑦):(𝐴𝐶)–1-1-onto→(𝐵𝐷)) → (𝐴𝐶) ≈ (𝐵𝐷))
96, 7, 8sylancr 590 . . . . . 6 (((𝑥:𝐴1-1-onto𝐵𝑦:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐴𝐶) ≈ (𝐵𝐷))
109ex 416 . . . . 5 ((𝑥:𝐴1-1-onto𝐵𝑦:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≈ (𝐵𝐷)))
1110exlimivv 1933 . . . 4 (∃𝑥𝑦(𝑥:𝐴1-1-onto𝐵𝑦:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≈ (𝐵𝐷)))
123, 11sylbir 238 . . 3 ((∃𝑥 𝑥:𝐴1-1-onto𝐵 ∧ ∃𝑦 𝑦:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≈ (𝐵𝐷)))
131, 2, 12syl2anb 600 . 2 ((𝐴𝐵𝐶𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≈ (𝐵𝐷)))
1413imp 410 1 (((𝐴𝐵𝐶𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐴𝐶) ≈ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  Vcvv 3444  cun 3882  cin 3883  c0 4246   class class class wbr 5033  1-1-ontowf1o 6327  cen 8493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-en 8497
This theorem is referenced by:  enpr2d  8584  difsnen  8586  undom  8592  limensuci  8681  infensuc  8683  phplem2  8685  pssnn  8724  dif1en  8739  unfi  8773  infdifsn  9108  pm54.43  9418  dif1card  9425  endjudisj  9583  djuen  9584  ssfin4  9725  fin23lem26  9740  unsnen  9968  fzennn  13335  mreexexlem4d  16914
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