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Mirrors > Home > MPE Home > Th. List > sselpwd | Structured version Visualization version GIF version |
Description: Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.) |
Ref | Expression |
---|---|
sselpwd.1 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
sselpwd.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
sselpwd | ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sselpwd.1 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
2 | sselpwd.2 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
3 | 1, 2 | ssexd 5220 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
4 | 3, 2 | elpwd 4548 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3495 ⊆ wss 3935 𝒫 cpw 4537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-in 3942 df-ss 3951 df-pw 4539 |
This theorem is referenced by: knatar 7099 marypha1 8887 fin1a2lem7 9817 canthp1lem2 10064 wunss 10123 ramub1lem1 16352 mreexd 16903 mreexexlemd 16905 mreexexlem4d 16908 opsrval 20185 selvfval 20260 cncls 21812 fbasrn 22422 rnelfmlem 22490 ustssel 22743 crefi 31011 ldsysgenld 31319 ldgenpisyslem1 31322 bj-ismoored 34294 bj-imdirval2 34366 rfovcnvf1od 40230 fsovrfovd 40235 fsovfd 40238 fsovcnvlem 40239 ntrclsrcomplex 40265 clsk3nimkb 40270 clsk1indlem4 40274 clsk1indlem1 40275 ntrclsiso 40297 ntrclskb 40299 ntrclsk3 40300 ntrclsk13 40301 ntrneircomplex 40304 ntrneik3 40326 ntrneix3 40327 ntrneik13 40328 ntrneix13 40329 clsneircomplex 40333 clsneiel1 40338 neicvgrcomplex 40343 neicvgel1 40349 mnussd 40479 mnuprssd 40485 mnuop3d 40487 wessf1ornlem 41325 ovolsplit 42154 |
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