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Theorem wunxp 10701
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 10687 . . . 4 (𝜑 → (𝐴𝐵) ∈ 𝑈)
51, 4wunpw 10684 . . 3 (𝜑 → 𝒫 (𝐴𝐵) ∈ 𝑈)
61, 5wunpw 10684 . 2 (𝜑 → 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈)
7 xpsspw 5801 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
87a1i 11 . 2 (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵))
91, 6, 8wunss 10689 1 (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cun 3942  wss 3944  𝒫 cpw 4596   × cxp 5667  WUnicwun 10677
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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-opab 5204  df-tr 5259  df-xp 5675  df-rel 5676  df-wun 10679
This theorem is referenced by:  wunpm  10702  wuncnv  10707  wunco  10710  wuntpos  10711  tskxp  10764  wuncn  11147  wunfunc  17831  wunfuncOLD  17832  wunnat  17889  wunnatOLD  17890  catcoppccl  18049  catcoppcclOLD  18050  catcfuccl  18051  catcfucclOLD  18052  catcxpccl  18141  catcxpcclOLD  18142
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