Theorem psspwb 42669

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

Assertion

Ref	Expression
psspwb	⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)

Proof of Theorem psspwb

Step	Hyp	Ref	Expression
1		sspwb 5401	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2		pweqb 5408	. . . 4 ⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
3	2	necon3bii 2984	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵)
4	1, 3	anbi12i 629	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
5		df-pss 3909	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
6		df-pss 3909	. 2 ⊢ (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
7	4, 5, 6	3bitr4i 303	1 ⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395   ≠ wne 2932   ⊆ wss 3889   ⊊ wpss 3890  𝒫 cpw 4541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3431  df-un 3894  df-ss 3906  df-pss 3909  df-pw 4543  df-sn 4568  df-pr 4570

This theorem is referenced by: (None)