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Theorem disjenex 9168
Description: Existence version of disjen 9167. (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 483 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
2 snex 5437 . . 3 {𝒫 ran 𝐴} ∈ V
3 xpexg 7760 . . 3 ((𝐵𝑊 ∧ {𝒫 ran 𝐴} ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
41, 2, 3sylancl 584 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
5 disjen 9167 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
6 ineq2 4208 . . . 4 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (𝐴𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
76eqeq1d 2730 . . 3 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → ((𝐴𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅))
8 breq1 5155 . . 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 3587 1 ((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3473  cin 3948  c0 4326  𝒫 cpw 4606  {csn 4632   cuni 4912   class class class wbr 5152   × cxp 5680  ran crn 5683  cen 8969
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1st 8001  df-2nd 8002  df-en 8973
This theorem is referenced by: (None)
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