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Theorem wunint 10753
Description: A weak universe is closed under nonempty intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunint ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)

Proof of Theorem wunint
StepHypRef Expression
1 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
21adantr 480 . 2 ((𝜑𝐴 ≠ ∅) → 𝑈 ∈ WUni)
3 wununi.2 . . . 4 (𝜑𝐴𝑈)
41, 3wununi 10744 . . 3 (𝜑 𝐴𝑈)
54adantr 480 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
6 intssuni 4975 . . 3 (𝐴 ≠ ∅ → 𝐴 𝐴)
76adantl 481 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴 𝐴)
82, 5, 7wunss 10750 1 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wne 2938  wss 3963  c0 4339   cuni 4912   cint 4951  WUnicwun 10738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-uni 4913  df-int 4952  df-tr 5266  df-wun 10740
This theorem is referenced by: (None)
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