Theorem disjenex 9174

Description: Existence version of disjen 9173. (Contributed by Mario Carneiro, 7-Feb-2015.)

Assertion

Ref	Expression
disjenex	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝑥,𝑊

Proof of Theorem disjenex

Step	Hyp	Ref	Expression
1		simpr 484	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
2		snex 5442	. . 3 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
3		xpexg 7769	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ {𝒫 ∪ ran 𝐴} ∈ V) → (𝐵 × {𝒫 ∪ ran 𝐴}) ∈ V)
4	1, 2, 3	sylancl 586	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝒫 ∪ ran 𝐴}) ∈ V)
5		disjen 9173	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))
6		ineq2 4222	. . . 4 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (𝐴 ∩ 𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})))
7	6	eqeq1d 2737	. . 3 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → ((𝐴 ∩ 𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅))
8		breq1 5151	. . 3 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (𝑥 ≈ 𝐵 ↔ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))
9	7, 8	anbi12d 632	. 2 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵) ↔ ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)))
10	4, 5, 9	spcedv 3598	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  Vcvv 3478   ∩ cin 3962  ∅c0 4339  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912   class class class wbr 5148   × cxp 5687  ran crn 5690   ≈ cen 8981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1st 8013  df-2nd 8014  df-en 8985

This theorem is referenced by: (None)