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Theorem endisj 9098
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 5307 . . . 4 ∅ ∈ V
31, 2xpsnen 9095 . . 3 (𝐴 × {∅}) ≈ 𝐴
4 endisj.2 . . . 4 𝐵 ∈ V
5 1oex 8516 . . . 4 1o ∈ V
64, 5xpsnen 9095 . . 3 (𝐵 × {1o}) ≈ 𝐵
73, 6pm3.2i 470 . 2 ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)
8 xp01disj 8529 . 2 ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅
9 p0ex 5384 . . . 4 {∅} ∈ V
101, 9xpex 7773 . . 3 (𝐴 × {∅}) ∈ V
11 snex 5436 . . . 4 {1o} ∈ V
124, 11xpex 7773 . . 3 (𝐵 × {1o}) ∈ V
13 breq1 5146 . . . . 5 (𝑥 = (𝐴 × {∅}) → (𝑥𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
14 breq1 5146 . . . . 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 4215 . . . . 5 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o})))
1716eqeq1d 2739 . . . 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 3607 . 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 1540  wex 1779  wcel 2108  Vcvv 3480  cin 3950  c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  1oc1o 8499  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-1o 8506  df-en 8986
This theorem is referenced by: (None)
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