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Theorem wunss 10750
Description: A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
wunss.3 (𝜑𝐵𝐴)
Assertion
Ref Expression
wunss (𝜑𝐵𝑈)

Proof of Theorem wunss
StepHypRef Expression
1 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
2 wununi.2 . . . 4 (𝜑𝐴𝑈)
31, 2wunpw 10745 . . 3 (𝜑 → 𝒫 𝐴𝑈)
41, 3wunelss 10746 . 2 (𝜑 → 𝒫 𝐴𝑈)
5 wunss.3 . . 3 (𝜑𝐵𝐴)
62, 5sselpwd 5334 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
74, 6sseldd 3996 1 (𝜑𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3963  𝒫 cpw 4605  WUnicwun 10738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607  df-uni 4913  df-tr 5266  df-wun 10740
This theorem is referenced by:  wunin  10751  wundif  10752  wunint  10753  wun0  10756  wunom  10758  wunxp  10762  wunpm  10763  wunmap  10764  wundm  10766  wunrn  10767  wuncnv  10768  wunres  10769  wunfv  10770  wunco  10771  wuntpos  10772  wuncn  11208  wunstr  17222  wunndx  17229  wunfunc  17952  wunfuncOLD  17953
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