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Theorem altxpexg 34017
Description: The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)
Assertion
Ref Expression
altxpexg ((𝐴𝑉𝐵𝑊) → (𝐴 ×× 𝐵) ∈ V)

Proof of Theorem altxpexg
StepHypRef Expression
1 altxpsspw 34016 . 2 (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
2 pwexg 5271 . . . 4 (𝐵𝑊 → 𝒫 𝐵 ∈ V)
3 unexg 7534 . . . 4 ((𝐴𝑉 ∧ 𝒫 𝐵 ∈ V) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
42, 3sylan2 596 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
5 pwexg 5271 . . 3 ((𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
6 pwexg 5271 . . 3 (𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
74, 5, 63syl 18 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
8 ssexg 5216 . 2 (((𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∧ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) → (𝐴 ×× 𝐵) ∈ V)
91, 7, 8sylancr 590 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ×× 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2110  Vcvv 3408  cun 3864  wss 3866  𝒫 cpw 4513   ×× caltxp 33996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-pw 4515  df-sn 4542  df-pr 4544  df-uni 4820  df-altop 33997  df-altxp 33998
This theorem is referenced by: (None)
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