Theorem unipwr

Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw . The proof of this theorem was automatically generated from unipwrVD 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
Assertion	unipwr	$$⊢A\subseteq \bigcup 𝒫A$$

Proof

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