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Theorem wunxp 10144
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 10130 . . . 4 (𝜑 → (𝐴𝐵) ∈ 𝑈)
51, 4wunpw 10127 . . 3 (𝜑 → 𝒫 (𝐴𝐵) ∈ 𝑈)
61, 5wunpw 10127 . 2 (𝜑 → 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈)
7 xpsspw 5669 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
87a1i 11 . 2 (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵))
91, 6, 8wunss 10132 1 (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  cun 3917  wss 3919  𝒫 cpw 4522   × cxp 5540  WUnicwun 10120
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-opab 5115  df-tr 5159  df-xp 5548  df-rel 5549  df-wun 10122
This theorem is referenced by:  wunpm  10145  wuncnv  10150  wunco  10153  wuntpos  10154  tskxp  10207  wuncn  10590  wunfunc  17169  wunnat  17226  catcoppccl  17368  catcfuccl  17369  catcxpccl  17457
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