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Theorem wunelss 10748
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 10745 . . 3 (𝑈 ∈ WUni → Tr 𝑈)
31, 2syl 17 . 2 (𝜑 → Tr 𝑈)
4 wununi.2 . 2 (𝜑𝐴𝑈)
5 trss 5270 . 2 (Tr 𝑈 → (𝐴𝑈𝐴𝑈))
63, 4, 5sylc 65 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951  Tr wtr 5259  WUnicwun 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260  df-wun 10742
This theorem is referenced by:  wunss  10752  wunf  10767  wuncval2  10787
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