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Theorem wunsn 10132
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 4563 . 2 {𝐴} = {𝐴, 𝐴}
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3, 3wunpr 10125 . 2 (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
51, 4eqeltrid 2920 1 (𝜑 → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  {csn 4550  {cpr 4552  WUnicwun 10116
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3015  df-ral 3138  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-uni 4826  df-tr 5160  df-wun 10118
This theorem is referenced by:  wunsuc  10133  wunfi  10137  wunop  10138  wuntpos  10150  wunsets  16522  1strwunbndx  16598  catcoppccl  17366
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