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Theorem endisj 9096
Description: Any two sets are equinumerous to two disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
Hypotheses
Ref Expression
endisj.1 𝐴 ∈ V
endisj.2 𝐵 ∈ V
Assertion
Ref Expression
endisj 𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem endisj
StepHypRef Expression
1 endisj.1 . . . 4 𝐴 ∈ V
2 0ex 5312 . . . 4 ∅ ∈ V
31, 2xpsnen 9093 . . 3 (𝐴 × {∅}) ≈ 𝐴
4 endisj.2 . . . 4 𝐵 ∈ V
5 1oex 8514 . . . 4 1o ∈ V
64, 5xpsnen 9093 . . 3 (𝐵 × {1o}) ≈ 𝐵
73, 6pm3.2i 470 . 2 ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)
8 xp01disj 8527 . 2 ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅
9 p0ex 5389 . . . 4 {∅} ∈ V
101, 9xpex 7771 . . 3 (𝐴 × {∅}) ∈ V
11 snex 5441 . . . 4 {1o} ∈ V
124, 11xpex 7771 . . 3 (𝐵 × {1o}) ∈ V
13 breq1 5150 . . . . 5 (𝑥 = (𝐴 × {∅}) → (𝑥𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
14 breq1 5150 . . . . 5 (𝑦 = (𝐵 × {1o}) → (𝑦𝐵 ↔ (𝐵 × {1o}) ≈ 𝐵))
1513, 14bi2anan9 638 . . . 4 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥𝐴𝑦𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)))
16 ineq12 4222 . . . . 5 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o})))
1716eqeq1d 2736 . . . 4 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅))
1815, 17anbi12d 632 . . 3 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅) ↔ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅)))
1910, 12, 18spc2ev 3606 . 2 ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅) → ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅))
207, 8, 19mp2an 692 1 𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wex 1775  wcel 2105  Vcvv 3477  cin 3961  c0 4338  {csn 4630   class class class wbr 5147   × cxp 5686  1oc1o 8497  cen 8980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-suc 6391  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-1o 8504  df-en 8984
This theorem is referenced by: (None)
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