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Theorem wunelss 9845
 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 9842 . . 3 (𝑈 ∈ WUni → Tr 𝑈)
31, 2syl 17 . 2 (𝜑 → Tr 𝑈)
4 wununi.2 . 2 (𝜑𝐴𝑈)
5 trss 4984 . 2 (Tr 𝑈 → (𝐴𝑈𝐴𝑈))
63, 4, 5sylc 65 1 (𝜑𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2164   ⊆ wss 3798  Tr wtr 4975  WUnicwun 9837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-v 3416  df-in 3805  df-ss 3812  df-uni 4659  df-tr 4976  df-wun 9839 This theorem is referenced by:  wunss  9849  wunf  9864  wuncval2  9884
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