Theorem endisj 9030

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 5254	. . . 4 ⊢ ∅ ∈ V
3	1, 2	xpsnen 9027	. . 3 ⊢ (𝐴 × {∅}) ≈ 𝐴
4		endisj.2	. . . 4 ⊢ 𝐵 ∈ V
5		1oex 8441	. . . 4 ⊢ 1o ∈ V
6	4, 5	xpsnen 9027	. . 3 ⊢ (𝐵 × {1o}) ≈ 𝐵
7	3, 6	pm3.2i 474	. 2 ⊢ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)
8		xp01disj 8454	. 2 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅
9		p0ex 5338	. . . 4 ⊢ {∅} ∈ V
10	1, 9	xpex 7731	. . 3 ⊢ (𝐴 × {∅}) ∈ V
11		snex 5393	. . . 4 ⊢ {1o} ∈ V
12	4, 11	xpex 7731	. . 3 ⊢ (𝐵 × {1o}) ∈ V
13		breq1 5100	. . . . 5 ⊢ (𝑥 = (𝐴 × {∅}) → (𝑥 ≈ 𝐴 ↔ (𝐴 × {∅}) ≈ 𝐴))
14		breq1 5100	. . . . 5 ⊢ (𝑦 = (𝐵 × {1o}) → (𝑦 ≈ 𝐵 ↔ (𝐵 × {1o}) ≈ 𝐵))
15	13, 14	bi2anan9 647	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵)))
16		ineq12 4165	. . . . 5 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → (𝑥 ∩ 𝑦) = ((𝐴 × {∅}) ∩ (𝐵 × {1o})))
17	16	eqeq1d 2763	. . . 4 ⊢ ((𝑥 = (𝐴 × {∅}) ∧ 𝑦 = (𝐵 × {1o})) → ((𝑥 ∩ 𝑦) = ∅ ↔ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅))
18	15, 17	anbi12d 641	. . 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 3565	. 2 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1o}) ≈ 𝐵) ∧ ((𝐴 × {∅}) ∩ (𝐵 × {1o})) = ∅) → ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅))
20	7, 8, 19	mp2an 702	1 ⊢ ∃𝑥∃𝑦((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ∧ (𝑥 ∩ 𝑦) = ∅)

Syntax hints:   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453   ∩ cin 3901  ∅c0 4283  {csn 4579   class class class wbr 5097   × cxp 5641  1_oc1o 8424   ≈ cen 8918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-suc 6347  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-1o 8431  df-en 8922

This theorem is referenced by: (None)