MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wunint Structured version   Visualization version   GIF version

Theorem wunint 10129
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 10120 . . 3 (𝜑 𝐴𝑈)
54adantr 481 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
6 intssuni 4895 . . 3 (𝐴 ≠ ∅ → 𝐴 𝐴)
76adantl 482 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴 𝐴)
82, 5, 7wunss 10126 1 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2107  wne 3020  wss 3939  c0 4294   cuni 4836   cint 4873  WUnicwun 10114
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-in 3946  df-ss 3955  df-nul 4295  df-pw 4543  df-uni 4837  df-int 4874  df-tr 5169  df-wun 10116
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator