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

Theorem psspwb 39221
 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 5316 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 pweqb 5323 . . . 4 (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
32necon3bii 3058 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵)
41, 3anbi12i 628 . 2 ((𝐴𝐵𝐴𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
5 df-pss 3930 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
6 df-pss 3930 . 2 (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
74, 5, 63bitr4i 305 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   ≠ wne 3006   ⊆ wss 3912   ⊊ wpss 3913  𝒫 cpw 4513 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pr 5304 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-v 3475  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-pss 3930  df-nul 4268  df-pw 4515  df-sn 4542  df-pr 4544 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator