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Mirrors > Home > MPE Home > Th. List > sspwd | Structured version Visualization version GIF version |
Description: The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.) |
Ref | Expression |
---|---|
sspwd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
sspwd | ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sspw 4616 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3963 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-pw 4607 |
This theorem is referenced by: pweq 4619 pwel 5387 marypha1lem 9471 pwwf 9845 rankpwi 9861 ackbij2lem1 10256 fictb 10282 ssfin2 10358 ssfin3ds 10368 ttukeylem2 10548 hashbcss 17038 isacs1i 17702 mreacs 17703 acsfn 17704 isacs3lem 18600 isacs5lem 18603 tgcmp 23425 imastopn 23744 fgabs 23903 fgtr 23914 trfg 23915 ssufl 23942 alexsubb 24070 cfiluweak 24320 cmetss 25364 minveclem4a 25478 minveclem4 25480 madess 27930 ldsysgenld 34141 neibastop1 36342 neibastop2lem 36343 neibastop2 36344 sstotbnd2 37761 prjcrv0 42620 isnacs3 42698 aomclem2 43044 sge0iunmptlemre 46371 |
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