Theorem grusn

Description: A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013)

	Ref	Expression
Assertion	grusn	⊢ ( ( 𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ) → { 𝐴 } ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		dfsn2	⊢ { 𝐴 } = { 𝐴 , 𝐴 }
2		grupr	⊢ ( ( 𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ) → { 𝐴 , 𝐴 } ∈ 𝑈 )
3	2	3anidm23	⊢ ( ( 𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ) → { 𝐴 , 𝐴 } ∈ 𝑈 )
4	1 3	eqeltrid	⊢ ( ( 𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ) → { 𝐴 } ∈ 𝑈 )