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Theorem endisj 8203
 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 4924 . . . 4 ∅ ∈ V
31, 2xpsnen 8200 . . 3 (𝐴 × {∅}) ≈ 𝐴
4 endisj.2 . . . 4 𝐵 ∈ V
5 1oex 7721 . . . 4 1𝑜 ∈ V
64, 5xpsnen 8200 . . 3 (𝐵 × {1𝑜}) ≈ 𝐵
73, 6pm3.2i 447 . 2 ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵)
8 xp01disj 7730 . 2 ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅
9 p0ex 4984 . . . 4 {∅} ∈ V
101, 9xpex 7109 . . 3 (𝐴 × {∅}) ∈ V
11 snex 5036 . . . 4 {1𝑜} ∈ V
124, 11xpex 7109 . . 3 (𝐵 × {1𝑜}) ∈ V
13 breq1 4789 . . . . 5 (𝑥 = (𝐴 × {∅}) → (𝑥𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
14 breq1 4789 . . . . 5 (𝑦 = (𝐵 × {1𝑜}) → (𝑦𝐵 ↔ (𝐵 × {1𝑜}) ≈ 𝐵))
1513, 14bi2anan9 620 . . . 4 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥𝐴𝑦𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵)))
16 ineq12 3960 . . . . 5 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → (𝑥𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})))
1716eqeq1d 2773 . . . 4 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅))
1815, 17anbi12d 616 . . 3 ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → (((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅) ↔ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅)))
1910, 12, 18spc2ev 3452 . 2 ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅) → ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅))
207, 8, 19mp2an 672 1 𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  Vcvv 3351   ∩ cin 3722  ∅c0 4063  {csn 4316   class class class wbr 4786   × cxp 5247  1𝑜c1o 7706   ≈ cen 8106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-ord 5869  df-on 5870  df-suc 5872  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-1o 7713  df-en 8110 This theorem is referenced by: (None)
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