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Theorem wunint 9650
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 472 . 2 ((𝜑𝐴 ≠ ∅) → 𝑈 ∈ WUni)
3 wununi.2 . . . 4 (𝜑𝐴𝑈)
41, 3wununi 9641 . . 3 (𝜑 𝐴𝑈)
54adantr 472 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
6 intssuni 4607 . . 3 (𝐴 ≠ ∅ → 𝐴 𝐴)
76adantl 473 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴 𝐴)
82, 5, 7wunss 9647 1 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2103  wne 2896  wss 3680  c0 4023   cuni 4544   cint 4583  WUnicwun 9635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-v 3306  df-dif 3683  df-in 3687  df-ss 3694  df-nul 4024  df-pw 4268  df-uni 4545  df-int 4584  df-tr 4861  df-wun 9637
This theorem is referenced by: (None)
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