Theorem altxpexg 35607

Description: The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)

Assertion

Ref	Expression
altxpexg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ×× 𝐵) ∈ V)

Proof of Theorem altxpexg

Step	Hyp	Ref	Expression
1		altxpsspw 35606	. 2 ⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
2		pwexg 5382	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ∈ V)
3		unexg 7757	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐵 ∈ V) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
4	2, 3	sylan2 591	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
5		pwexg 5382	. . 3 ⊢ ((𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
6		pwexg 5382	. . 3 ⊢ (𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
7	4, 5, 6	3syl 18	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
8		ssexg 5327	. 2 ⊢ (((𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∧ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) → (𝐴 ×× 𝐵) ∈ V)
9	1, 7, 8	sylancr 585	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ×× 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098  Vcvv 3473   ∪ cun 3947   ⊆ wss 3949  𝒫 cpw 4606   ×× caltxp 35586

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-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  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-pw 4608  df-sn 4633  df-pr 4635  df-uni 4913  df-altop 35587  df-altxp 35588

This theorem is referenced by: (None)