Theorem psspwb

Description: Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022)

	Ref	Expression
Assertion	psspwb	$$⊢A\subset B\leftrightarrow 𝒫A\subset 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		sspwb	$$⊢A\subseteq B\leftrightarrow 𝒫A\subseteq 𝒫B$$
2		pweqb	$$⊢A=B\leftrightarrow 𝒫A=𝒫B$$
3	2	necon3bii	$$⊢A\ne B\leftrightarrow 𝒫A\ne 𝒫B$$
4	1 3	anbi12i	$$⊢\left(A\subseteq B\wedge A\ne B\right)\leftrightarrow \left(𝒫A\subseteq 𝒫B\wedge 𝒫A\ne 𝒫B\right)$$
5		df-pss	$$⊢A\subset B\leftrightarrow \left(A\subseteq B\wedge A\ne B\right)$$
6		df-pss	$$⊢𝒫A\subset 𝒫B\leftrightarrow \left(𝒫A\subseteq 𝒫B\wedge 𝒫A\ne 𝒫B\right)$$
7	4 5 6	3bitr4i	$$⊢A\subset B\leftrightarrow 𝒫A\subset 𝒫B$$