Theorem disjenex 9059

Description: Existence version of disjen 9058. (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 5378	. . 3 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
3		xpexg 7692	. . 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 9058	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))
6		ineq2 4163	. . . 4 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (𝐴 ∩ 𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})))
7	6	eqeq1d 2735	. . 3 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → ((𝐴 ∩ 𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅))
8		breq1 5098	. . 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 3549	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897  ∅c0 4282  𝒫 cpw 4551  {csn 4577  ∪ cuni 4860   class class class wbr 5095   × cxp 5619  ran crn 5622   ≈ 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-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-1st 7930  df-2nd 7931  df-en 8880

This theorem is referenced by: (None)