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Theorem wunsn 10707
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 4640 . 2 {𝐴} = {𝐴, 𝐴}
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3, 3wunpr 10700 . 2 (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
51, 4eqeltrid 2837 1 (𝜑 → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  {csn 4627  {cpr 4629  WUnicwun 10691
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-un 3952  df-in 3954  df-ss 3964  df-sn 4628  df-pr 4630  df-uni 4908  df-tr 5265  df-wun 10693
This theorem is referenced by:  wunsuc  10708  wunfi  10712  wunop  10713  wuntpos  10725  wunsets  17106  1strwunbndx  17159  catcoppccl  18063  catcoppcclOLD  18064
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