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Theorem wunun 9645
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 4558 . . 3 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
41, 2, 3syl2anc 696 . 2 (𝜑 {𝐴, 𝐵} = (𝐴𝐵))
5 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
65, 1, 2wunpr 9644 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
75, 6wununi 9641 . 2 (𝜑 {𝐴, 𝐵} ∈ 𝑈)
84, 7eqeltrrd 2804 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  cun 3678  {cpr 4287   cuni 4544  WUnicwun 9635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-v 3306  df-un 3685  df-in 3687  df-ss 3694  df-sn 4286  df-pr 4288  df-uni 4545  df-tr 4861  df-wun 9637
This theorem is referenced by:  wuntp  9646  wunsuc  9652  wunfi  9656  wunxp  9659  wuntpos  9669  wunsets  16023  catcoppccl  16880
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