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Theorem wunss 10122
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 10117 . . 3 (𝜑 → 𝒫 𝐴𝑈)
41, 3wunelss 10118 . 2 (𝜑 → 𝒫 𝐴𝑈)
5 wunss.3 . . 3 (𝜑𝐵𝐴)
62, 5sselpwd 5221 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
74, 6sseldd 3965 1 (𝜑𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wss 3933  𝒫 cpw 4535  WUnicwun 10110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949  df-pw 4537  df-uni 4831  df-tr 5164  df-wun 10112
This theorem is referenced by:  wunin  10123  wundif  10124  wunint  10125  wun0  10128  wunom  10130  wunxp  10134  wunpm  10135  wunmap  10136  wundm  10138  wunrn  10139  wuncnv  10140  wunres  10141  wunfv  10142  wunco  10143  wuntpos  10144  wuncn  10580  wunndx  16492  wunstr  16495  wunfunc  17157
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