Theorem psspwb 37928

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 5073	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2		pweqb 5081	. . . 4 ⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
3	2	necon3bii 2989	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵)
4	1, 3	anbi12i 620	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
5		df-pss 3748	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
6		df-pss 3748	. 2 ⊢ (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
7	4, 5, 6	3bitr4i 294	1 ⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)

Syntax hints:   ↔ wb 197   ∧ wa 384   ≠ wne 2937   ⊆ wss 3732   ⊊ wpss 3733  𝒫 cpw 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-pw 4317  df-sn 4335  df-pr 4337

This theorem is referenced by: (None)