Theorem psspwb 43375

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 5416	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2		pweqb 5423	. . . 4 ⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
3	2	necon3bii 3007	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵)
4	1, 3	anbi12i 640	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
5		df-pss 4014	. 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵))
6		df-pss 4014	. 2 ⊢ (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
7	4, 5, 6	3bitr4i 306	1 ⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ≠ wne 2955   ⊆ wss 3895   ⊊ wpss 4013  𝒫 cpw 4556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-ss 3896  df-pss 4014  df-un 4151  df-pw 4558  df-sn 4585  df-pr 4587

This theorem is used by: (None)