Theorem endisj 7932

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	⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem endisj

Step	Hyp	Ref	Expression
1		endisj.1	. . . 4 ⊢ 𝐴 ∈ V
2		0ex 4718	. . . 4 ⊢ ∅ ∈ V
3	1, 2	xpsnen 7929	. . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴
4		endisj.2	. . . 4 ⊢ 𝐵 ∈ V
5		1on 7454	. . . . 5 ⊢ 1𝑜 ∈ On
6	5	elexi 3186	. . . 4 ⊢ 1𝑜 ∈ V
7	4, 6	xpsnen 7929	. . 3 ⊢ (𝐵 × {1𝑜}) ≈ 𝐵
8	3, 7	pm3.2i 470	. 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵)
9		xp01disj 7463	. 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅
10		p0ex 4779	. . . 4 ⊢ {∅} ∈ V
11	1, 10	xpex 6860	. . 3 ⊢ (𝐴 × {∅}) ∈ V
12		snex 4835	. . . 4 ⊢ {1𝑜} ∈ V
13	4, 12	xpex 6860	. . 3 ⊢ (𝐵 × {1𝑜}) ∈ V
14		breq1 4586	. . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
15		breq1 4586	. . . . 5 ⊢ (𝑦 = (𝐵 × {1𝑜}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1𝑜}) ≈ 𝐵))
16	14, 15	bi2anan9 913	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵)))
17		ineq12 3771	. . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})))
18	17	eqeq1d 2612	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅))
19	16, 18	anbi12d 743	. . 3 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → (((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅)))
20	11, 13, 19	spc2ev 3274	. 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅))
21	8, 9, 20	mp2an 704	1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ∅c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  Oncon0 5640  1_𝑜c1o 7440   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-ord 5643  df-on 5644  df-suc 5646  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-1o 7447  df-en 7842

This theorem is referenced by: (None)