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Theorem wunss 9820
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 9815 . . 3 (𝜑 → 𝒫 𝐴𝑈)
41, 3wunelss 9816 . 2 (𝜑 → 𝒫 𝐴𝑈)
5 wunss.3 . . 3 (𝜑𝐵𝐴)
62, 5sselpwd 5000 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
74, 6sseldd 3797 1 (𝜑𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  wss 3767  𝒫 cpw 4347  WUnicwun 9808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775  ax-sep 4973
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-v 3385  df-in 3774  df-ss 3781  df-pw 4349  df-uni 4627  df-tr 4944  df-wun 9810
This theorem is referenced by:  wunin  9821  wundif  9822  wunint  9823  wun0  9826  wunom  9828  wunxp  9832  wunpm  9833  wunmap  9834  wundm  9836  wunrn  9837  wuncnv  9838  wunres  9839  wunfv  9840  wunco  9841  wuntpos  9842  wuncn  10277  wunndx  16202  wunstr  16205  wunfunc  16870
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