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Mirrors > Home > MPE Home > Th. List > sspwi | Structured version Visualization version GIF version |
Description: The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.) |
Ref | Expression |
---|---|
sspwi.1 | ⊢ 𝐴 ⊆ 𝐵 |
Ref | Expression |
---|---|
sspwi | ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwi.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | sspw 4545 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊆ 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: pwunss 4552 pwundif 4558 pwdom 8662 wdompwdom 9035 rankxplim 9301 hashbclem 13807 incexclem 15184 sscpwex 17078 wunfunc 17162 tsmsres 22745 cfilresi 23891 vitali 24207 sqff1o 25755 ldgenpisyslem1 31441 imambfm 31539 ballotlem2 31765 dssmapnvod 40447 gneispace 40565 sge0less 42755 |
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