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Theorem wunelss 10119
Description: The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunelss (𝜑𝐴𝑈)

Proof of Theorem wunelss
StepHypRef Expression
1 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
2 wuntr 10116 . . 3 (𝑈 ∈ WUni → Tr 𝑈)
31, 2syl 17 . 2 (𝜑 → Tr 𝑈)
4 wununi.2 . 2 (𝜑𝐴𝑈)
5 trss 5157 . 2 (Tr 𝑈 → (𝐴𝑈𝐴𝑈))
63, 4, 5sylc 65 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  wss 3908  Tr wtr 5148  WUnicwun 10111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ne 3012  df-ral 3135  df-v 3471  df-in 3915  df-ss 3925  df-uni 4814  df-tr 5149  df-wun 10113
This theorem is referenced by:  wunss  10123  wunf  10138  wuncval2  10158
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