Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  wunsuc Structured version   Visualization version   GIF version

Theorem wunsuc 9937
 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 6035 . 2 suc 𝐴 = (𝐴 ∪ {𝐴})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3wunsn 9936 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
52, 3, 4wunun 9930 . 2 (𝜑 → (𝐴 ∪ {𝐴}) ∈ 𝑈)
61, 5syl5eqel 2870 1 (𝜑 → suc 𝐴𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2050   ∪ cun 3827  {csn 4441  suc csuc 6031  WUnicwun 9920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-v 3417  df-un 3834  df-in 3836  df-ss 3843  df-sn 4442  df-pr 4444  df-uni 4713  df-tr 5031  df-suc 6035  df-wun 9922 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator