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Theorem wunun 10750
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 4923 . . 3 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
41, 2, 3syl2anc 584 . 2 (𝜑 {𝐴, 𝐵} = (𝐴𝐵))
5 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
65, 1, 2wunpr 10749 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
75, 6wununi 10746 . 2 (𝜑 {𝐴, 𝐵} ∈ 𝑈)
84, 7eqeltrrd 2842 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cun 3949  {cpr 4628   cuni 4907  WUnicwun 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908  df-tr 5260  df-wun 10742
This theorem is referenced by:  wuntp  10751  wunsuc  10757  wunfi  10761  wunxp  10764  wuntpos  10774  wunsets  17214  catcoppccl  18162
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