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Theorem wunin 9850
 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 4057 . . 3 (𝐴𝐵) ⊆ 𝐴
43a1i 11 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
51, 2, 4wunss 9849 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2164   ∩ cin 3797   ⊆ wss 3798  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  ax-sep 5005 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-pw 4380  df-uni 4659  df-tr 4976  df-wun 9839 This theorem is referenced by:  wunress  16304
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