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Theorem wunxp 10718
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 10704 . . . 4 (𝜑 → (𝐴𝐵) ∈ 𝑈)
51, 4wunpw 10701 . . 3 (𝜑 → 𝒫 (𝐴𝐵) ∈ 𝑈)
61, 5wunpw 10701 . 2 (𝜑 → 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈)
7 xpsspw 5809 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
87a1i 11 . 2 (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵))
91, 6, 8wunss 10706 1 (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cun 3946  wss 3948  𝒫 cpw 4602   × cxp 5674  WUnicwun 10694
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-opab 5211  df-tr 5266  df-xp 5682  df-rel 5683  df-wun 10696
This theorem is referenced by:  wunpm  10719  wuncnv  10724  wunco  10727  wuntpos  10728  tskxp  10781  wuncn  11164  wunfunc  17848  wunfuncOLD  17849  wunnat  17906  wunnatOLD  17907  catcoppccl  18066  catcoppcclOLD  18067  catcfuccl  18068  catcfucclOLD  18069  catcxpccl  18158  catcxpcclOLD  18159
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