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Theorem altxpexg 34553
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 34552 . 2 (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
2 pwexg 5333 . . . 4 (𝐵𝑊 → 𝒫 𝐵 ∈ V)
3 unexg 7682 . . . 4 ((𝐴𝑉 ∧ 𝒫 𝐵 ∈ V) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
42, 3sylan2 593 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
5 pwexg 5333 . . 3 ((𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
6 pwexg 5333 . . 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 5280 . 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 396  wcel 2106  Vcvv 3445  cun 3908  wss 3910  𝒫 cpw 4560   ×× caltxp 34532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-pw 4562  df-sn 4587  df-pr 4589  df-uni 4866  df-altop 34533  df-altxp 34534
This theorem is referenced by: (None)
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