Theorem endisj 8988

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	⊢ ∃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 5249	. . . 4 ⊢ ∅ ∈ V
3	1, 2	xpsnen 8985	. . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴
4		endisj.2	. . . 4 ⊢ 𝐵 ∈ V
5		1oex 8404	. . . 4 ⊢ 1o ∈ V
6	4, 5	xpsnen 8985	. . 3 ⊢ (𝐵 × {1o}) ≈ 𝐵
7	3, 6	pm3.2i 470	. 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)
8		xp01disj 8415	. 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅
9		p0ex 5326	. . . 4 ⊢ {∅} ∈ V
10	1, 9	xpex 7695	. . 3 ⊢ (𝐴 × {∅}) ∈ V
11		snex 5378	. . . 4 ⊢ {1o} ∈ V
12	4, 11	xpex 7695	. . 3 ⊢ (𝐵 × {1o}) ∈ V
13		breq1 5098	. . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
14		breq1 5098	. . . . 5 ⊢ (𝑦 = (𝐵 × {1o}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1o}) ≈ 𝐵))
15	13, 14	bi2anan9 638	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)))
16		ineq12 4164	. . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o})))
17	16	eqeq1d 2735	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅))
18	15, 17	anbi12d 632	. . 3 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅) ↔ (((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅)))
19	10, 12, 18	spc2ev 3558	. 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅))
20	7, 8, 19	mp2an 692	1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897  ∅c0 4282  {csn 4577   class class class wbr 5095   × cxp 5619  1_oc1o 8387   ≈ cen 8876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-suc 6320  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-1o 8394  df-en 8880

This theorem is referenced by: (None)