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Theorem altxpexg 35960
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 35959 . 2 (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
2 pwexg 5384 . . . 4 (𝐵𝑊 → 𝒫 𝐵 ∈ V)
3 unexg 7762 . . . 4 ((𝐴𝑉 ∧ 𝒫 𝐵 ∈ V) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
42, 3sylan2 593 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
5 pwexg 5384 . . 3 ((𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
6 pwexg 5384 . . 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 5329 . 2 (((𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∧ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) → (𝐴 ×× 𝐵) ∈ V)
91, 7, 8sylancr 587 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ×× 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  Vcvv 3478  cun 3961  wss 3963  𝒫 cpw 4605   ×× caltxp 35939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-altop 35940  df-altxp 35941
This theorem is referenced by: (None)
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