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Mirrors > Home > MPE Home > Th. List > Mathboxes > psspwb | Structured version Visualization version GIF version |
Description: Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.) |
Ref | Expression |
---|---|
psspwb | ⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwb 5445 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) | |
2 | pweqb 5452 | . . . 4 ⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵) | |
3 | 2 | necon3bii 2983 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵) |
4 | 1, 3 | anbi12i 626 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵)) |
5 | df-pss 3966 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
6 | df-pss 3966 | . 2 ⊢ (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵)) | |
7 | 4, 5, 6 | 3bitr4i 302 | 1 ⊢ (𝐴 ⊊ 𝐵 ↔ 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) |
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