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 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sspwb	⊢ ( 𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵 )
2		pweqb	⊢ ( 𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵 )
3	2	necon3bii	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵 )
4	1 3	anbi12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) ↔ ( 𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵 ) )
5		df-pss	⊢ ( 𝐴 ⊊ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) )
6		df-pss	⊢ ( 𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ ( 𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵 ) )
7	4 5 6	3bitr4i	⊢ ( 𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵 )