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Theorem wunin 10650
Description: A weak universe is closed under binary intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunin (𝜑 → (𝐴𝐵) ∈ 𝑈)

Proof of Theorem wunin
StepHypRef Expression
1 wununi.1 . 2 (𝜑𝑈 ∈ WUni)
2 wununi.2 . 2 (𝜑𝐴𝑈)
3 inss1 4189 . . 3 (𝐴𝐵) ⊆ 𝐴
43a1i 11 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
51, 2, 4wunss 10649 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cin 3910  wss 3911  WUnicwun 10637
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 2708  ax-sep 5257
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rab 3409  df-v 3448  df-in 3918  df-ss 3928  df-pw 4563  df-uni 4867  df-tr 5224  df-wun 10639
This theorem is referenced by:  wunress  17132  wunressOLD  17133
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