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Theorem altxpexg 34609
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 34608 . 2 (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
2 pwexg 5334 . . . 4 (𝐵𝑊 → 𝒫 𝐵 ∈ V)
3 unexg 7684 . . . 4 ((𝐴𝑉 ∧ 𝒫 𝐵 ∈ V) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
42, 3sylan2 594 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
5 pwexg 5334 . . 3 ((𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
6 pwexg 5334 . . 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 5281 . 2 (((𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∧ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) → (𝐴 ×× 𝐵) ∈ V)
91, 7, 8sylancr 588 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ×× 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Vcvv 3444  cun 3909  wss 3911  𝒫 cpw 4561   ×× caltxp 34588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-pw 4563  df-sn 4588  df-pr 4590  df-uni 4867  df-altop 34589  df-altxp 34590
This theorem is referenced by: (None)
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