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Theorem disjenex 8003
 Description: Existence version of disjen 8002. (Contributed by Mario Carneiro, 7-Feb-2015.)
Assertion
Ref Expression
disjenex ((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝑥,𝑊

Proof of Theorem disjenex
StepHypRef Expression
1 simpr 476 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
2 snex 4835 . . 3 {𝒫 ran 𝐴} ∈ V
3 xpexg 6858 . . 3 ((𝐵𝑊 ∧ {𝒫 ran 𝐴} ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
41, 2, 3sylancl 693 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
5 disjen 8002 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
6 ineq2 3770 . . . . 5 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (𝐴𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
76eqeq1d 2612 . . . 4 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → ((𝐴𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅))
8 breq1 4586 . . . 4 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (𝑥𝐵 ↔ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
97, 8anbi12d 743 . . 3 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (((𝐴𝑥) = ∅ ∧ 𝑥𝐵) ↔ ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)))
109spcegv 3267 . 2 ((𝐵 × {𝒫 ran 𝐴}) ∈ V → (((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵)))
114, 5, 10sylc 63 1 ((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  ran crn 5039   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1st 7059  df-2nd 7060  df-en 7842 This theorem is referenced by: (None)
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