Theorem altxpexg 36368

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 36367	. 2 ⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
2		pwexg 5350	. . . 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 5350	. . 3 ⊢ ((𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
6		pwexg 5350	. . 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 5294	. 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 2149  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  𝒫 cpw 4567   ×× caltxp 36347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rex 3096  df-rab 3424  df-v 3465  df-un 3918  df-in 3920  df-ss 3930  df-pw 4569  df-sn 4595  df-pr 4597  df-uni 4877  df-altop 36348  df-altxp 36349

This theorem is referenced by: (None)