Theorem snsslVD

Description: Virtual deduction proof of snssl . (Contributed by Alan Sare, 25-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		snsslVD.1	$$⊢A\in \mathrm{V}$$
2		idn1	$$⊢\left(\left\{A\right\}\subseteq B\to \left\{A\right\}\subseteq B\right)$$
3	1	snid	$$⊢A\in \left\{A\right\}$$
4		ssel2	$$⊢\left(\left\{A\right\}\subseteq B\wedge A\in \left\{A\right\}\right)\to A\in B$$
5	2 3 4	e10an	$$⊢\left(\left\{A\right\}\subseteq B\to A\in B\right)$$
6	5	in1	$$⊢\left\{A\right\}\subseteq B\to A\in B$$