Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  psspwb Structured version   Visualization version   GIF version

Theorem psspwb 41969
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 (𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)

Proof of Theorem psspwb
StepHypRef Expression
1 sspwb 5445 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 pweqb 5452 . . . 4 (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
32necon3bii 2983 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵)
41, 3anbi12i 626 . 2 ((𝐴𝐵𝐴𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
5 df-pss 3966 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
6 df-pss 3966 . 2 (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
74, 5, 63bitr4i 302 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wne 2930  wss 3946  wpss 3947  𝒫 cpw 4597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5294  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-v 3464  df-un 3951  df-ss 3963  df-pss 3966  df-pw 4599  df-sn 4624  df-pr 4626
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator