Theorem wunsn 9417

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 4138	. 2 ⊢ {𝐴} = {𝐴, 𝐴}
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		wununi.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	2, 3, 3	wunpr 9410	. 2 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
5	1, 4	syl5eqel 2692	1 ⊢ (𝜑 → {𝐴} ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 1977  {csn 4125  {cpr 4127  WUnicwun 9401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128  df-uni 4373  df-tr 4681  df-wun 9403

This theorem is referenced by:  wunsuc  9418  wunfi  9422  wunop  9423  wuntpos  9435  wunsets  15728  1strwunbndx  15807  catcoppccl  16581