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Theorem wunxp 10715
Description: A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
wunop.3 (𝜑𝐵𝑈)
Assertion
Ref Expression
wunxp (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)

Proof of Theorem wunxp
StepHypRef Expression
1 wun0.1 . 2 (𝜑𝑈 ∈ WUni)
2 wunop.2 . . . . 5 (𝜑𝐴𝑈)
3 wunop.3 . . . . 5 (𝜑𝐵𝑈)
41, 2, 3wunun 10701 . . . 4 (𝜑 → (𝐴𝐵) ∈ 𝑈)
51, 4wunpw 10698 . . 3 (𝜑 → 𝒫 (𝐴𝐵) ∈ 𝑈)
61, 5wunpw 10698 . 2 (𝜑 → 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈)
7 xpsspw 5799 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
87a1i 11 . 2 (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵))
91, 6, 8wunss 10703 1 (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  cun 3938  wss 3940  𝒫 cpw 4594   × cxp 5664  WUnicwun 10691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-opab 5201  df-tr 5256  df-xp 5672  df-rel 5673  df-wun 10693
This theorem is referenced by:  wunpm  10716  wuncnv  10721  wunco  10724  wuntpos  10725  tskxp  10778  wuncn  11161  wunfunc  17850  wunfuncOLD  17851  wunnat  17909  wunnatOLD  17910  catcoppccl  18069  catcoppcclOLD  18070  catcfuccl  18071  catcfucclOLD  18072  catcxpccl  18161  catcxpcclOLD  18162
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