Theorem unipwrVD

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

	Ref	Expression
Assertion	unipwrVD	$$⊢A\subseteq \bigcup 𝒫A$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2	1	snid	$$⊢x\in \left\{x\right\}$$
3		idn1	$$⊢\left(x\in A\to x\in A\right)$$
4		snelpwi	$$⊢x\in A\to \left\{x\right\}\in 𝒫A$$
5	3 4	e1a	$$⊢\left(x\in A\to \left\{x\right\}\in 𝒫A\right)$$
6		elunii	$$⊢\left(x\in \left\{x\right\}\wedge \left\{x\right\}\in 𝒫A\right)\to x\in \bigcup 𝒫A$$
7	2 5 6	e01an	$$⊢\left(x\in A\to x\in \bigcup 𝒫A\right)$$
8	7	in1	$$⊢x\in A\to x\in \bigcup 𝒫A$$
9	8	ssriv	$$⊢A\subseteq \bigcup 𝒫A$$