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Theorem endisj 9057
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 5300 . . . 4 ∅ ∈ V
31, 2xpsnen 9054 . . 3 (𝐴 × {∅}) ≈ 𝐴
4 endisj.2 . . . 4 𝐵 ∈ V
5 1oex 8474 . . . 4 1o ∈ V
64, 5xpsnen 9054 . . 3 (𝐵 × {1o}) ≈ 𝐵
73, 6pm3.2i 470 . 2 ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)
8 xp01disj 8489 . 2 ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅
9 p0ex 5375 . . . 4 {∅} ∈ V
101, 9xpex 7736 . . 3 (𝐴 × {∅}) ∈ V
11 snex 5424 . . . 4 {1o} ∈ V
124, 11xpex 7736 . . 3 (𝐵 × {1o}) ∈ V
13 breq1 5144 . . . . 5 (𝑥 = (𝐴 × {∅}) → (𝑥𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
14 breq1 5144 . . . . 5 (𝑦 = (𝐵 × {1o}) → (𝑦𝐵 ↔ (𝐵 × {1o}) ≈ 𝐵))
1513, 14bi2anan9 636 . . . 4 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥𝐴𝑦𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)))
16 ineq12 4202 . . . . 5 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o})))
1716eqeq1d 2728 . . . 4 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅))
1815, 17anbi12d 630 . . 3 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅) ↔ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅)))
1910, 12, 18spc2ev 3591 . 2 ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅) → ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅))
207, 8, 19mp2an 689 1 𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1533  wex 1773  wcel 2098  Vcvv 3468  cin 3942  c0 4317  {csn 4623   class class class wbr 5141   × cxp 5667  1oc1o 8457  cen 8935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-suc 6363  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-1o 8464  df-en 8939
This theorem is referenced by: (None)
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