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

Theorem wunsn 9875
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 4411 . 2 {𝐴} = {𝐴, 𝐴}
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3, 3wunpr 9868 . 2 (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
51, 4syl5eqel 2863 1 (𝜑 → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  {csn 4398  {cpr 4400  WUnicwun 9859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-v 3400  df-un 3797  df-in 3799  df-ss 3806  df-sn 4399  df-pr 4401  df-uni 4674  df-tr 4990  df-wun 9861
This theorem is referenced by:  wunsuc  9876  wunfi  9880  wunop  9881  wuntpos  9893  wunsets  16307  1strwunbndx  16384  catcoppccl  17154
  Copyright terms: Public domain W3C validator