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

Theorem wunun 10126
 Description: A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
wunpr.3 (𝜑𝐵𝑈)
Assertion
Ref Expression
wunun (𝜑 → (𝐴𝐵) ∈ 𝑈)

Proof of Theorem wunun
StepHypRef Expression
1 wununi.2 . . 3 (𝜑𝐴𝑈)
2 wunpr.3 . . 3 (𝜑𝐵𝑈)
3 uniprg 4843 . . 3 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
41, 2, 3syl2anc 587 . 2 (𝜑 {𝐴, 𝐵} = (𝐴𝐵))
5 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
65, 1, 2wunpr 10125 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
75, 6wununi 10122 . 2 (𝜑 {𝐴, 𝐵} ∈ 𝑈)
84, 7eqeltrrd 2917 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115   ∪ cun 3917  {cpr 4552  ∪ cuni 4825  WUnicwun 10116 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3015  df-ral 3138  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-uni 4826  df-tr 5160  df-wun 10118 This theorem is referenced by:  wuntp  10127  wunsuc  10133  wunfi  10137  wunxp  10140  wuntpos  10150  wunsets  16522  catcoppccl  17366
 Copyright terms: Public domain W3C validator