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

Theorem wunint 10740
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 479 . 2 ((𝜑𝐴 ≠ ∅) → 𝑈 ∈ WUni)
3 wununi.2 . . . 4 (𝜑𝐴𝑈)
41, 3wununi 10731 . . 3 (𝜑 𝐴𝑈)
54adantr 479 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
6 intssuni 4974 . . 3 (𝐴 ≠ ∅ → 𝐴 𝐴)
76adantl 480 . 2 ((𝜑𝐴 ≠ ∅) → 𝐴 𝐴)
82, 5, 7wunss 10737 1 ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  wne 2929  wss 3944  c0 4322   cuni 4909   cint 4950  WUnicwun 10725
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 2696  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4323  df-pw 4606  df-uni 4910  df-int 4951  df-tr 5267  df-wun 10727
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator