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

Theorem wunsn 9738
Description: A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
Assertion
Ref Expression
wunsn (𝜑 → {𝐴} ∈ 𝑈)

Proof of Theorem wunsn
StepHypRef Expression
1 dfsn2 4329 . 2 {𝐴} = {𝐴, 𝐴}
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3, 3wunpr 9731 . 2 (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
51, 4syl5eqel 2854 1 (𝜑 → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  {csn 4316  {cpr 4318  WUnicwun 9722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-un 3728  df-in 3730  df-ss 3737  df-sn 4317  df-pr 4319  df-uni 4575  df-tr 4887  df-wun 9724
This theorem is referenced by:  wunsuc  9739  wunfi  9743  wunop  9744  wuntpos  9756  wunsets  16100  1strwunbndx  16182  catcoppccl  16958
  Copyright terms: Public domain W3C validator