Theorem endisj 7051

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 4221	. . . 4 ⊢ ∅ ∈ V
3	1, 2	xpsnen 7048	. . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴
4		endisj.2	. . . 4 ⊢ 𝐵 ∈ V
5		1on 6632	. . . . 5 ⊢ 1o ∈ On
6	5	elexi 2816	. . . 4 ⊢ 1o ∈ V
7	4, 6	xpsnen 7048	. . 3 ⊢ (𝐵 × {1o}) ≈ 𝐵
8	3, 7	pm3.2i 272	. 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)
9		xp01disj 6644	. 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅
10		p0ex 4284	. . . 4 ⊢ {∅} ∈ V
11	1, 10	xpex 4848	. . 3 ⊢ (𝐴 × {∅}) ∈ V
12	6	snex 4281	. . . 4 ⊢ {1o} ∈ V
13	4, 12	xpex 4848	. . 3 ⊢ (𝐵 × {1o}) ∈ V
14		breq1 4096	. . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
15		breq1 4096	. . . . 5 ⊢ (𝑦 = (𝐵 × {1o}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1o}) ≈ 𝐵))
16	14, 15	bi2anan9 610	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)))
17		ineq12 3405	. . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o})))
18	17	eqeq1d 2240	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅))
19	16, 18	anbi12d 473	. . 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 2903	. 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅))
21	8, 9, 20	mp2an 426	1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803   ∩ cin 3200  ∅c0 3496  {csn 3673   class class class wbr 4093  Oncon0 4466   × cxp 4729  1_oc1o 6618   ≈ cen 6950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-1o 6625  df-en 6953

This theorem is referenced by: (None)