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Metamath Proof Explorer

Theorem snss

Description: The singleton of an element of a class is a subset of the class (inference form of snssg ). Theorem 7.4 of Quine p. 49. (Contributed by NM, 21-Jun-1993)

		Ref	Expression
	Hypothesis	snss.1	⊢ 𝐴 ∈ V
	Assertion	snss	⊢ ( 𝐴 ∈ 𝐵 ↔ { 𝐴 } ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		snss.1	⊢ 𝐴 ∈ V
2		snssg	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝐵 ↔ { 𝐴 } ⊆ 𝐵 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ 𝐵 ↔ { 𝐴 } ⊆ 𝐵 )