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Theorem wunelss 9490
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 9487 . . 3 (𝑈 ∈ WUni → Tr 𝑈)
31, 2syl 17 . 2 (𝜑 → Tr 𝑈)
4 wununi.2 . 2 (𝜑𝐴𝑈)
5 trss 4731 . 2 (Tr 𝑈 → (𝐴𝑈𝐴𝑈))
63, 4, 5sylc 65 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3560  Tr wtr 4722  WUnicwun 9482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-v 3192  df-in 3567  df-ss 3574  df-uni 4410  df-tr 4723  df-wun 9484
This theorem is referenced by:  wunss  9494  wunf  9509  wuncval2  9529
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