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Theorem wunelss 10703
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 10700 . . 3 (𝑈 ∈ WUni → Tr 𝑈)
31, 2syl 17 . 2 (𝜑 → Tr 𝑈)
4 wununi.2 . 2 (𝜑𝐴𝑈)
5 trss 5277 . 2 (Tr 𝑈 → (𝐴𝑈𝐴𝑈))
63, 4, 5sylc 65 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wss 3949  Tr wtr 5266  WUnicwun 10695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-v 3477  df-in 3956  df-ss 3966  df-uni 4910  df-tr 5267  df-wun 10697
This theorem is referenced by:  wunss  10707  wunf  10722  wuncval2  10742
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