Theorem wunsn 10627

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 4593	. 2 ⊢ {𝐴} = {𝐴, 𝐴}
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		wununi.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	2, 3, 3	wunpr 10620	. 2 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
5	1, 4	eqeltrid 2840	1 ⊢ (𝜑 → {𝐴} ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2113  {csn 4580  {cpr 4582  WUnicwun 10611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-v 3442  df-un 3906  df-ss 3918  df-sn 4581  df-pr 4583  df-uni 4864  df-tr 5206  df-wun 10613

This theorem is referenced by:  wunsuc  10628  wunfi  10632  wunop  10633  wuntpos  10645  wunsets  17104  1strwunbndx  17152  catcoppccl  18041