Theorem sspwimpcf

Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of TakeutiZaring p. 18. sspwimpcf , using conventional notation, was translated from its virtual deduction form, sspwimpcfVD , using a translation program. (Contributed by Alan Sare, 13-Jun-2015) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2		id	$$⊢A\subseteq B\to A\subseteq B$$
3		id	$$⊢x\in 𝒫A\to x\in 𝒫A$$
4		elpwi	$$⊢x\in 𝒫A\to x\subseteq A$$
5	3 4	syl	$$⊢x\in 𝒫A\to x\subseteq A$$
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	syl2an	$$⊢\left(A\subseteq B\wedge x\in 𝒫A\right)\to x\subseteq B$$
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	eel021old	$$⊢\left(A\subseteq B\wedge x\in 𝒫A\right)\to x\in 𝒫B$$
12	11	ex	$$⊢A\subseteq B\to \left(x\in 𝒫A\to x\in 𝒫B\right)$$
13	12	alrimiv	$$⊢A\subseteq B\to \forall x\left(x\in 𝒫A\to x\in 𝒫B\right)$$
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	syl	$$⊢A\subseteq B\to 𝒫A\subseteq 𝒫B$$
17	16	iin1	$$⊢A\subseteq B\to 𝒫A\subseteq 𝒫B$$