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	$$⊢A\in \mathrm{V}$$
Assertion	snss	$$⊢A\in B\leftrightarrow \left\{A\right\}\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		snss.1	$$⊢A\in \mathrm{V}$$
2		snssg	$$⊢A\in \mathrm{V}\to \left(A\in B\leftrightarrow \left\{A\right\}\subseteq B\right)$$
3	1 2	ax-mp	$$⊢A\in B\leftrightarrow \left\{A\right\}\subseteq B$$