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Theorem psspwb 39769
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 5305 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 pweqb 5312 . . . 4 (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
32necon3bii 2986 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵)
41, 3anbi12i 630 . 2 ((𝐴𝐵𝐴𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
5 df-pss 3860 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
6 df-pss 3860 . 2 (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
74, 5, 63bitr4i 306 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wne 2934  wss 3841  wpss 3842  𝒫 cpw 4485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-pw 4487  df-sn 4514  df-pr 4516
This theorem is referenced by: (None)
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