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Theorem psspwb 40203
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 5365 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 pweqb 5372 . . . 4 (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
32necon3bii 2996 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵)
41, 3anbi12i 627 . 2 ((𝐴𝐵𝐴𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
5 df-pss 3906 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
6 df-pss 3906 . 2 (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
74, 5, 63bitr4i 303 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wne 2943  wss 3887  wpss 3888  𝒫 cpw 4533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564
This theorem is referenced by: (None)
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