Theorem snssl

Description: If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss . The proof of this theorem was automatically generated from snsslVD using a tools command file, translateMWO.cmd, by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	snssl.1	$$⊢A\in \mathrm{V}$$
Assertion	snssl	$$⊢\left\{A\right\}\subseteq B\to A\in B$$

Proof

Step	Hyp	Ref	Expression
1		snssl.1	$$⊢A\in \mathrm{V}$$
2	1	snid	$$⊢A\in \left\{A\right\}$$
3		ssel2	$$⊢\left(\left\{A\right\}\subseteq B\wedge A\in \left\{A\right\}\right)\to A\in B$$
4	2 3	mpan2	$$⊢\left\{A\right\}\subseteq B\to A\in B$$