Theorem sspwimpcfVD

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 ) using conjunction-form virtual hypothesis collections. It was completed automatically by a tools program which would invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimpcf is sspwimpcfVD without virtual deductions and was derived from sspwimpcfVD . The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sspwimpcfVD	$$⊢A\subseteq B\to 𝒫A\subseteq 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2		idn1	$$⊢\left(A\subseteq B\to A\subseteq B\right)$$
3		idn1	$$⊢\left(x\in 𝒫A\to x\in 𝒫A\right)$$
4		elpwi	$$⊢x\in 𝒫A\to x\subseteq A$$
5	3 4	el1	$$⊢\left(x\in 𝒫A\to x\subseteq A\right)$$
6		sstr2	$$⊢x\subseteq A\to \left(A\subseteq B\to x\subseteq B\right)$$
7	6	impcom	$$⊢\left(A\subseteq B\wedge x\subseteq A\right)\to x\subseteq B$$
8	2 5 7	el12	$$⊢\left(\left(A\subseteq B,x\in 𝒫A\right)\to x\subseteq B\right)$$
9		elpwg	$$⊢x\in \mathrm{V}\to \left(x\in 𝒫B\leftrightarrow x\subseteq B\right)$$
10	9	biimpar	$$⊢\left(x\in \mathrm{V}\wedge x\subseteq B\right)\to x\in 𝒫B$$
11	1 8 10	el021old	$$⊢\left(\left(A\subseteq B,x\in 𝒫A\right)\to x\in 𝒫B\right)$$
12	11	int2	$$⊢\left(A\subseteq B\to \left(x\in 𝒫A\to x\in 𝒫B\right)\right)$$
13	12	gen11	$$⊢\left(A\subseteq B\to \forall x\left(x\in 𝒫A\to x\in 𝒫B\right)\right)$$
14		dfss2	$$⊢𝒫A\subseteq 𝒫B\leftrightarrow \forall x\left(x\in 𝒫A\to x\in 𝒫B\right)$$
15	14	biimpri	$$⊢\forall x\left(x\in 𝒫A\to x\in 𝒫B\right)\to 𝒫A\subseteq 𝒫B$$
16	13 15	el1	$$⊢\left(A\subseteq B\to 𝒫A\subseteq 𝒫B\right)$$
17	16	in1	$$⊢A\subseteq B\to 𝒫A\subseteq 𝒫B$$