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Theorem wunun 10725
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 4919 . . 3 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
41, 2, 3syl2anc 583 . 2 (𝜑 {𝐴, 𝐵} = (𝐴𝐵))
5 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
65, 1, 2wunpr 10724 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
75, 6wununi 10721 . 2 (𝜑 {𝐴, 𝐵} ∈ 𝑈)
84, 7eqeltrrd 2829 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cun 3942  {cpr 4626   cuni 4903  WUnicwun 10715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-v 3471  df-un 3949  df-in 3951  df-ss 3961  df-sn 4625  df-pr 4627  df-uni 4904  df-tr 5260  df-wun 10717
This theorem is referenced by:  wuntp  10726  wunsuc  10732  wunfi  10736  wunxp  10739  wuntpos  10749  wunsets  17137  catcoppccl  18097  catcoppcclOLD  18098
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