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Theorem wunxp 9490
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 9476 . . . 4 (𝜑 → (𝐴𝐵) ∈ 𝑈)
51, 4wunpw 9473 . . 3 (𝜑 → 𝒫 (𝐴𝐵) ∈ 𝑈)
61, 5wunpw 9473 . 2 (𝜑 → 𝒫 𝒫 (𝐴𝐵) ∈ 𝑈)
7 xpsspw 5194 . . 3 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
87a1i 11 . 2 (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵))
91, 6, 8wunss 9478 1 (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cun 3553  wss 3555  𝒫 cpw 4130   × cxp 5072  WUnicwun 9466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-opab 4674  df-tr 4713  df-xp 5080  df-rel 5081  df-wun 9468
This theorem is referenced by:  wunpm  9491  wuncnv  9496  wunco  9499  wuntpos  9500  tskxp  9553  wuncn  9935  wunfunc  16480  wunnat  16537  catcoppccl  16679  catcfuccl  16680  catcxpccl  16768
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