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Theorem wunsuc 9752
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 5891 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3wunsn 9751 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
52, 3, 4wunun 9745 . 2 (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
61, 5syl5eqel 2844 1 (𝜑 → suc 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2140  cun 3714  {csn 4322  suc csuc 5887  WUnicwun 9735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-v 3343  df-un 3721  df-in 3723  df-ss 3730  df-sn 4323  df-pr 4325  df-uni 4590  df-tr 4906  df-suc 5891  df-wun 9737
This theorem is referenced by: (None)
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