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Theorem wunss 10133
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 10128 . . 3 (𝜑 → 𝒫 𝐴𝑈)
41, 3wunelss 10129 . 2 (𝜑 → 𝒫 𝐴𝑈)
5 wunss.3 . . 3 (𝜑𝐵𝐴)
62, 5sselpwd 5229 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
74, 6sseldd 3967 1 (𝜑𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  wss 3935  𝒫 cpw 4538  WUnicwun 10121
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3496  df-in 3942  df-ss 3951  df-pw 4540  df-uni 4838  df-tr 5172  df-wun 10123
This theorem is referenced by:  wunin  10134  wundif  10135  wunint  10136  wun0  10139  wunom  10141  wunxp  10145  wunpm  10146  wunmap  10147  wundm  10149  wunrn  10150  wuncnv  10151  wunres  10152  wunfv  10153  wunco  10154  wuntpos  10155  wuncn  10591  wunndx  16503  wunstr  16506  wunfunc  17168
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