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| Mirrors > Home > MPE Home > Th. List > pwidg | Structured version Visualization version GIF version | ||
| Description: A set is an element of its power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| pwidg | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssid 4005 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
| 2 | elpwg 4602 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴)) | |
| 3 | 1, 2 | mpbiri 258 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ss 3967 df-pw 4601 | 
| This theorem is referenced by: pwidb 4620 pwid 4621 axpweq 5350 knatar 7378 pwssfi 9218 brwdom2 9614 pwwf 9848 rankpwi 9864 canthp1lem2 10694 canthp1 10695 mremre 17648 submre 17649 baspartn 22962 fctop 23012 cctop 23014 ppttop 23015 epttop 23017 isopn3 23075 mretopd 23101 tsmsfbas 24137 exsslsb 33648 gsumesum 34061 esumcst 34065 pwsiga 34132 prsiga 34133 sigainb 34138 pwldsys 34159 ldgenpisyslem1 34165 carsggect 34321 ex-sategoelel 35427 neibastop1 36361 neibastop2lem 36362 topdifinfindis 37348 elrfi 42710 dssmapnvod 44038 ntrk0kbimka 44057 clsk3nimkb 44058 neik0pk1imk0 44065 ntrclscls00 44084 ntrneicls00 44107 dvnprodlem3 45968 caragenunidm 46528 | 
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