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Theorem disjenex 8698
 Description: Existence version of disjen 8697. (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 489 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
2 snex 5301 . . 3 {𝒫 ran 𝐴} ∈ V
3 xpexg 7472 . . 3 ((𝐵𝑊 ∧ {𝒫 ran 𝐴} ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
41, 2, 3sylancl 590 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
5 disjen 8697 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
6 ineq2 4112 . . . 4 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (𝐴𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
76eqeq1d 2761 . . 3 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → ((𝐴𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅))
8 breq1 5036 . . 3 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (𝑥𝐵 ↔ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
97, 8anbi12d 634 . 2 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (((𝐴𝑥) = ∅ ∧ 𝑥𝐵) ↔ ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)))
104, 5, 9spcedv 3518 1 ((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539  ∃wex 1782   ∈ wcel 2112  Vcvv 3410   ∩ cin 3858  ∅c0 4226  𝒫 cpw 4495  {csn 4523  ∪ cuni 4799   class class class wbr 5033   × cxp 5523  ran crn 5526   ≈ cen 8525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-int 4840  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-1st 7694  df-2nd 7695  df-en 8529 This theorem is referenced by: (None)
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