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Theorem disjenex 8471
Description: Existence version of disjen 8470. (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 477 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
2 snex 5188 . . 3 {𝒫 ran 𝐴} ∈ V
3 xpexg 7290 . . 3 ((𝐵𝑊 ∧ {𝒫 ran 𝐴} ∈ V) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
41, 2, 3sylancl 577 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝒫 ran 𝐴}) ∈ V)
5 disjen 8470 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
6 ineq2 4070 . . . 4 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (𝐴𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})))
76eqeq1d 2780 . . 3 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → ((𝐴𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅))
8 breq1 4932 . . 3 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (𝑥𝐵 ↔ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))
97, 8anbi12d 621 . 2 (𝑥 = (𝐵 × {𝒫 ran 𝐴}) → (((𝐴𝑥) = ∅ ∧ 𝑥𝐵) ↔ ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵)))
104, 5, 9elabd 3583 1 ((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wex 1742  wcel 2050  Vcvv 3415  cin 3828  c0 4178  𝒫 cpw 4422  {csn 4441   cuni 4712   class class class wbr 4929   × cxp 5405  ran crn 5408  cen 8303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-int 4750  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-1st 7501  df-2nd 7502  df-en 8307
This theorem is referenced by: (None)
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