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Theorem wunin 10705
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 4221 . . 3 (𝐴𝐵) ⊆ 𝐴
43a1i 11 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
51, 2, 4wunss 10704 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  cin 3940  wss 3941  WUnicwun 10692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-in 3948  df-ss 3958  df-pw 4597  df-uni 4901  df-tr 5257  df-wun 10694
This theorem is referenced by:  wunress  17200  wunressOLD  17201
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