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Theorem endisj 7991
Description: Any two sets are equinumerous to 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 4750 . . . 4 ∅ ∈ V
31, 2xpsnen 7988 . . 3 (𝐴 × {∅}) ≈ 𝐴
4 endisj.2 . . . 4 𝐵 ∈ V
5 1on 7512 . . . . 5 1𝑜 ∈ On
65elexi 3199 . . . 4 1𝑜 ∈ V
74, 6xpsnen 7988 . . 3 (𝐵 × {1𝑜}) ≈ 𝐵
83, 7pm3.2i 471 . 2 ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵)
9 xp01disj 7521 . 2 ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅
10 p0ex 4813 . . . 4 {∅} ∈ V
111, 10xpex 6915 . . 3 (𝐴 × {∅}) ∈ V
12 snex 4869 . . . 4 {1𝑜} ∈ V
134, 12xpex 6915 . . 3 (𝐵 × {1𝑜}) ∈ V
14 breq1 4616 . . . . 5 (𝑥 = (𝐴 × {∅}) → (𝑥𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
15 breq1 4616 . . . . 5 (𝑦 = (𝐵 × {1𝑜}) → (𝑦𝐵 ↔ (𝐵 × {1𝑜}) ≈ 𝐵))
1614, 15bi2anan9 916 . . . 4 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥𝐴𝑦𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵)))
17 ineq12 3787 . . . . 5 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → (𝑥𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})))
1817eqeq1d 2623 . . . 4 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅))
1916, 18anbi12d 746 . . 3 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅) ↔ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅)))
2011, 13, 19spc2ev 3287 . 2 ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅) → ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅))
218, 9, 20mp2an 707 1 𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cin 3554  c0 3891  {csn 4148   class class class wbr 4613   × cxp 5072  Oncon0 5682  1𝑜c1o 7498  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-ord 5685  df-on 5686  df-suc 5688  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-1o 7505  df-en 7900
This theorem is referenced by: (None)
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