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Theorem unen 6229
 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 (((AB 𝐶𝐷) ((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 6164 . . 3 (ABx x:A1-1-ontoB)
2 bren 6164 . . 3 (𝐶𝐷y y:𝐶1-1-onto𝐷)
3 eeanv 1804 . . . 4 (xy(x:A1-1-ontoB y:𝐶1-1-onto𝐷) ↔ (x x:A1-1-ontoB y y:𝐶1-1-onto𝐷))
4 vex 2554 . . . . . . . 8 x V
5 vex 2554 . . . . . . . 8 y V
64, 5unex 4142 . . . . . . 7 (xy) V
7 f1oun 5089 . . . . . . 7 (((x:A1-1-ontoB y:𝐶1-1-onto𝐷) ((A𝐶) = ∅ (B𝐷) = ∅)) → (xy):(A𝐶)–1-1-onto→(B𝐷))
8 f1oen3g 6170 . . . . . . 7 (((xy) V (xy):(A𝐶)–1-1-onto→(B𝐷)) → (A𝐶) ≈ (B𝐷))
96, 7, 8sylancr 393 . . . . . 6 (((x:A1-1-ontoB y:𝐶1-1-onto𝐷) ((A𝐶) = ∅ (B𝐷) = ∅)) → (A𝐶) ≈ (B𝐷))
109ex 108 . . . . 5 ((x:A1-1-ontoB y:𝐶1-1-onto𝐷) → (((A𝐶) = ∅ (B𝐷) = ∅) → (A𝐶) ≈ (B𝐷)))
1110exlimivv 1773 . . . 4 (xy(x:A1-1-ontoB y:𝐶1-1-onto𝐷) → (((A𝐶) = ∅ (B𝐷) = ∅) → (A𝐶) ≈ (B𝐷)))
123, 11sylbir 125 . . 3 ((x x:A1-1-ontoB y y:𝐶1-1-onto𝐷) → (((A𝐶) = ∅ (B𝐷) = ∅) → (A𝐶) ≈ (B𝐷)))
131, 2, 12syl2anb 275 . 2 ((AB 𝐶𝐷) → (((A𝐶) = ∅ (B𝐷) = ∅) → (A𝐶) ≈ (B𝐷)))
1413imp 115 1 (((AB 𝐶𝐷) ((A𝐶) = ∅ (B𝐷) = ∅)) → (A𝐶) ≈ (B𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   ∪ cun 2909   ∩ cin 2910  ∅c0 3218   class class class wbr 3755  –1-1-onto→wf1o 4844   ≈ cen 6155 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-en 6158 This theorem is referenced by:  frecfzennn  8884
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