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Theorem wunss 10752
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 10747 . . 3 (𝜑 → 𝒫 𝐴𝑈)
41, 3wunelss 10748 . 2 (𝜑 → 𝒫 𝐴𝑈)
5 wunss.3 . . 3 (𝜑𝐵𝐴)
62, 5sselpwd 5328 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
74, 6sseldd 3984 1 (𝜑𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951  𝒫 cpw 4600  WUnicwun 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602  df-uni 4908  df-tr 5260  df-wun 10742
This theorem is referenced by:  wunin  10753  wundif  10754  wunint  10755  wun0  10758  wunom  10760  wunxp  10764  wunpm  10765  wunmap  10766  wundm  10768  wunrn  10769  wuncnv  10770  wunres  10771  wunfv  10772  wunco  10773  wuntpos  10774  wuncn  11210  wunstr  17225  wunndx  17232  wunfunc  17946
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