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Theorem wunsuc 10132
 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 6169 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3wunsn 10131 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
52, 3, 4wunun 10125 . 2 (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
61, 5eqeltrid 2897 1 (𝜑 → suc 𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112   ∪ cun 3882  {csn 4528  suc csuc 6165  WUnicwun 10115 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-ral 3114  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-uni 4804  df-tr 5140  df-suc 6169  df-wun 10117 This theorem is referenced by: (None)
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