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Theorem psspwb 37928
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 5073 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 pweqb 5081 . . . 4 (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
32necon3bii 2989 . . 3 (𝐴𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵)
41, 3anbi12i 620 . 2 ((𝐴𝐵𝐴𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
5 df-pss 3748 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
6 df-pss 3748 . 2 (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵))
74, 5, 63bitr4i 294 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wne 2937  wss 3732  wpss 3733  𝒫 cpw 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-pw 4317  df-sn 4335  df-pr 4337
This theorem is referenced by: (None)
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