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Theorem wunsuc 10755
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 6392 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3wunsn 10754 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
52, 3, 4wunun 10748 . 2 (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
61, 5eqeltrid 2843 1 (𝜑 → suc 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cun 3961  {csn 4631  suc csuc 6388  WUnicwun 10738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-tr 5266  df-suc 6392  df-wun 10740
This theorem is referenced by: (None)
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