Theorem endisj 8047

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 4790	. . . 4 ⊢ ∅ ∈ V
3	1, 2	xpsnen 8044	. . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴
4		endisj.2	. . . 4 ⊢ 𝐵 ∈ V
5		1on 7567	. . . . 5 ⊢ 1𝑜 ∈ On
6	5	elexi 3213	. . . 4 ⊢ 1𝑜 ∈ V
7	4, 6	xpsnen 8044	. . 3 ⊢ (𝐵 × {1𝑜}) ≈ 𝐵
8	3, 7	pm3.2i 471	. 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵)
9		xp01disj 7576	. 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅
10		p0ex 4853	. . . 4 ⊢ {∅} ∈ V
11	1, 10	xpex 6962	. . 3 ⊢ (𝐴 × {∅}) ∈ V
12		snex 4908	. . . 4 ⊢ {1𝑜} ∈ V
13	4, 12	xpex 6962	. . 3 ⊢ (𝐵 × {1𝑜}) ∈ V
14		breq1 4656	. . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
15		breq1 4656	. . . . 5 ⊢ (𝑦 = (𝐵 × {1𝑜}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1𝑜}) ≈ 𝐵))
16	14, 15	bi2anan9 917	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵)))
17		ineq12 3809	. . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})))
18	17	eqeq1d 2624	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1𝑜})) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅))
19	16, 18	anbi12d 747	. . 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 3301	. 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅))
21	8, 9, 20	mp2an 708	1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)

Syntax hints:   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  Oncon0 5723  1_𝑜c1o 7553   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-ord 5726  df-on 5727  df-suc 5729  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-1o 7560  df-en 7956

This theorem is referenced by: (None)