Theorem disjenex 9122

Description: Existence version of disjen 9121. (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 489	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
2		snex 5411	. . 3 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
3		xpexg 7748	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ {𝒫 ∪ ran 𝐴} ∈ V) → (𝐵 × {𝒫 ∪ ran 𝐴}) ∈ V)
4	1, 2, 3	sylancl 597	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝒫 ∪ ran 𝐴}) ∈ V)
5		disjen 9121	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))
6		ineq2 4175	. . . 4 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (𝐴 ∩ 𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})))
7	6	eqeq1d 2771	. . 3 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → ((𝐴 ∩ 𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅))
8		breq1 5116	. . 3 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (𝑥 ≈ 𝐵 ↔ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))
9	7, 8	anbi12d 643	. 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 3566	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912  ∅c0 4294  𝒫 cpw 4567  {csn 4594  ∪ cuni 4876   class class class wbr 5113   × cxp 5660  ran crn 5663   ≈ cen 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7985  df-2nd 7986  df-en 8943

This theorem is referenced by: (None)