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Theorem wunun 10642
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 4880 . . 3 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
41, 2, 3syl2anc 584 . 2 (𝜑 {𝐴, 𝐵} = (𝐴𝐵))
5 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
65, 1, 2wunpr 10641 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
75, 6wununi 10638 . 2 (𝜑 {𝐴, 𝐵} ∈ 𝑈)
84, 7eqeltrrd 2839 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cun 3906  {cpr 4586   cuni 4863  WUnicwun 10632
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-v 3445  df-un 3913  df-in 3915  df-ss 3925  df-sn 4585  df-pr 4587  df-uni 4864  df-tr 5221  df-wun 10634
This theorem is referenced by:  wuntp  10643  wunsuc  10649  wunfi  10653  wunxp  10656  wuntpos  10666  wunsets  17041  catcoppccl  17995  catcoppcclOLD  17996
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