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Theorem wunsn 10756
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 4639 . 2 {𝐴} = {𝐴, 𝐴}
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wununi.2 . . 3 (𝜑𝐴𝑈)
42, 3, 3wunpr 10749 . 2 (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
51, 4eqeltrid 2845 1 (𝜑 → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  {csn 4626  {cpr 4628  WUnicwun 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908  df-tr 5260  df-wun 10742
This theorem is referenced by:  wunsuc  10757  wunfi  10761  wunop  10762  wuntpos  10774  wunsets  17214  1strwunbndx  17265  catcoppccl  18162
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