Theorem disjenex 9162

Description: Existence version of disjen 9161. (Contributed by Mario Carneiro, 7-Feb-2015.)

Assertion

Ref	Expression
disjenex	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉   𝑥,𝑊

Proof of Theorem disjenex

Step	Hyp	Ref	Expression
1		simpr 490	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
2		snex 5397	. . 3 ⊢ {𝒫 ∪ ran 𝐴} ∈ V
3		xpexg 7768	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ {𝒫 ∪ ran 𝐴} ∈ V) → (𝐵 × {𝒫 ∪ ran 𝐴}) ∈ V)
4	1, 2, 3	sylancl 598	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝒫 ∪ ran 𝐴}) ∈ V)
5		disjen 9161	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))
6		ineq2 4110	. . . 4 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (𝐴 ∩ 𝑥) = (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})))
7	6	eqeq1d 2762	. . 3 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → ((𝐴 ∩ 𝑥) = ∅ ↔ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅))
8		breq1 5105	. . 3 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (𝑥 ≈ 𝐵 ↔ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))
9	7, 8	anbi12d 644	. 2 ⊢ (𝑥 = (𝐵 × {𝒫 ∪ ran 𝐴}) → (((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵) ↔ ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)))
10	4, 5, 9	spcedv 3552	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450   ∩ cin 4088  ∅c0 4277  𝒫 cpw 4556  {csn 4584  ∪ cuni 4866   class class class wbr 5102   × cxp 5649  ran crn 5652   ≈ cen 8978

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 8005  df-2nd 8006  df-en 8982

This theorem is used by: (None)