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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 5345 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) | |
2 | pweqb 5352 | . . . 4 ⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵) | |
3 | 2 | necon3bii 3071 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝒫 𝐴 ≠ 𝒫 𝐵) |
4 | 1, 3 | anbi12i 628 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵)) |
5 | df-pss 3957 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
6 | df-pss 3957 | . 2 ⊢ (𝒫 𝐴 ⊊ 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐴 ≠ 𝒫 𝐵)) | |
7 | 4, 5, 6 | 3bitr4i 305 | 1 ⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ≠ wne 3019 ⊆ wss 3939 ⊊ wpss 3940 𝒫 cpw 4542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-pw 4544 df-sn 4571 df-pr 4573 |
This theorem is referenced by: (None) |
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