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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 4545 | . 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 3929 𝒫 cpw 4532 |
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 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-in 3936 df-ss 3945 df-pw 4534 |
This theorem is referenced by: pweq 4548 pwel 5275 marypha1lem 8890 pwwf 9229 rankpwi 9245 ackbij2lem1 9634 fictb 9660 ssfin2 9735 ssfin3ds 9745 ttukeylem2 9925 hashbcss 16333 isacs1i 16921 mreacs 16922 acsfn 16923 isacs3lem 17769 isacs5lem 17772 tgcmp 22002 imastopn 22321 fgabs 22480 fgtr 22491 trfg 22492 ssufl 22519 alexsubb 22647 cfiluweak 22897 cmetss 23912 minveclem4a 24026 minveclem4 24028 ldsysgenld 31438 neibastop1 33726 neibastop2lem 33727 neibastop2 33728 sstotbnd2 35086 isnacs3 39384 aomclem2 39732 sge0iunmptlemre 42772 |
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