Theorem wunsn 10708

Description: A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wununi.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wununi.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
wunsn	⊢ (𝜑 → {𝐴} ∈ 𝑈)

Proof of Theorem wunsn

Step	Hyp	Ref	Expression
1		dfsn2 4634	. 2 ⊢ {𝐴} = {𝐴, 𝐴}
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		wununi.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	2, 3, 3	wunpr 10701	. 2 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
5	1, 4	eqeltrid 2829	1 ⊢ (𝜑 → {𝐴} ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2098  {csn 4621  {cpr 4623  WUnicwun 10692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-v 3468  df-un 3946  df-in 3948  df-ss 3958  df-sn 4622  df-pr 4624  df-uni 4901  df-tr 5257  df-wun 10694

This theorem is referenced by:  wunsuc  10709  wunfi  10713  wunop  10714  wuntpos  10726  wunsets  17115  1strwunbndx  17168  catcoppccl  18075  catcoppcclOLD  18076