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Theorem wunsuc 10661
Description: A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunsuc (𝜑 → suc 𝐴𝑈)

Proof of Theorem wunsuc
StepHypRef Expression
1 df-suc 6327 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3wunsn 10660 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
52, 3, 4wunun 10654 . 2 (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
61, 5eqeltrid 2838 1 (𝜑 → suc 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cun 3912  {csn 4590  suc csuc 6323  WUnicwun 10644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-v 3449  df-un 3919  df-in 3921  df-ss 3931  df-sn 4591  df-pr 4593  df-uni 4870  df-tr 5227  df-suc 6327  df-wun 10646
This theorem is referenced by: (None)
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