Theorem altxpexg 36470

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 36469	. 2 ⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
2		pwexg 5349	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ∈ V)
3		unexg 7741	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐵 ∈ V) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
4	2, 3	sylan2 604	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
5		pwexg 5349	. . 3 ⊢ ((𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
6		pwexg 5349	. . 3 ⊢ (𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
7	4, 5, 6	3syl 19	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
8		ssexg 5290	. 2 ⊢ (((𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∧ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) → (𝐴 ×× 𝐵) ∈ V)
9	1, 7, 8	sylancr 598	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ×× 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  𝒫 cpw 4562   ×× caltxp 36449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rex 3090  df-rab 3417  df-v 3457  df-un 3910  df-in 3912  df-ss 3922  df-pw 4564  df-sn 4590  df-pr 4592  df-uni 4873  df-altop 36450  df-altxp 36451

This theorem is referenced by: (None)