Theorem snelpwrVD

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

	Ref	Expression
Assertion	snelpwrVD	$$⊢A\in B\to \left\{A\right\}\in 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		snex	$$⊢\left\{A\right\}\in \mathrm{V}$$
2		idn1	$$⊢\left(A\in B\to A\in B\right)$$
3		snssi	$$⊢A\in B\to \left\{A\right\}\subseteq B$$
4	2 3	e1a	$$⊢\left(A\in B\to \left\{A\right\}\subseteq B\right)$$
5		elpwg	$$⊢\left\{A\right\}\in \mathrm{V}\to \left(\left\{A\right\}\in 𝒫B\leftrightarrow \left\{A\right\}\subseteq B\right)$$
6	5	biimprd	$$⊢\left\{A\right\}\in \mathrm{V}\to \left(\left\{A\right\}\subseteq B\to \left\{A\right\}\in 𝒫B\right)$$
7	1 4 6	e01	$$⊢\left(A\in B\to \left\{A\right\}\in 𝒫B\right)$$
8	7	in1	$$⊢A\in B\to \left\{A\right\}\in 𝒫B$$