Theorem wunsn

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

	Ref	Expression
Hypotheses	wununi.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wununi.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion	wunsn	⊢ ( 𝜑 → { 𝐴 } ∈ 𝑈 )

Proof

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