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Theorem disjenex 9137
Description: Existence version of disjen 9136. (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 484 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
2 snex 5424 . . 3 {𝒫 ran 𝐴} ∈ V
3 xpexg 7734 . . 3 ((𝐵𝑊 ∧ {𝒫 ran 𝐴} ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
41, 2, 3sylancl 585 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
5 disjen 9136 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
6 ineq2 4201 . . . 4 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (𝐴𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
76eqeq1d 2728 . . 3 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → ((𝐴𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅))
8 breq1 5144 . . 3 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (𝑥𝐵 ↔ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
97, 8anbi12d 630 . 2 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (((𝐴𝑥) = ∅ ∧ 𝑥𝐵) ↔ ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)))
104, 5, 9spcedv 3582 1 ((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  Vcvv 3468  cin 3942  c0 4317  𝒫 cpw 4597  {csn 4623   cuni 4902   class class class wbr 5141   × cxp 5667  ran crn 5670  cen 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7974  df-2nd 7975  df-en 8942
This theorem is referenced by: (None)
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