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Theorem wunint 10572
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 481 . 2 ((𝜑𝐴 ≠ ∅) → 𝑈 ∈ WUni)
3 wununi.2 . . . 4 (𝜑𝐴𝑈)
41, 3wununi 10563 . . 3 (𝜑 𝐴𝑈)
54adantr 481 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
6 intssuni 4918 . . 3 (𝐴 ≠ ∅ → 𝐴 𝐴)
76adantl 482 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴 𝐴)
82, 5, 7wunss 10569 1 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wne 2940  wss 3898  c0 4269   cuni 4852   cint 4894  WUnicwun 10557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-in 3905  df-ss 3915  df-nul 4270  df-pw 4549  df-uni 4853  df-int 4895  df-tr 5210  df-wun 10559
This theorem is referenced by: (None)
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