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Mirrors > Home > MPE Home > Th. List > p0ex | Structured version Visualization version GIF version |
Description: The power set of the empty set (the ordinal 1) is a set. See also p0exALT 5390. (Contributed by NM, 23-Dec-1993.) |
Ref | Expression |
---|---|
p0ex | ⊢ {∅} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw0 4816 | . 2 ⊢ 𝒫 ∅ = {∅} | |
2 | 0ex 5312 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | pwex 5385 | . 2 ⊢ 𝒫 ∅ ∈ V |
4 | 1, 3 | eqeltrri 2835 | 1 ⊢ {∅} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Vcvv 3477 ∅c0 4338 𝒫 cpw 4604 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-dif 3965 df-ss 3979 df-nul 4339 df-pw 4606 df-sn 4631 |
This theorem is referenced by: pp0ex 5391 dtruALT 5393 zfpair 5426 tposexg 8263 fsetexb 8902 endisj 9096 pw2eng 9116 dfac4 10159 dfac2b 10168 axcc2lem 10473 axdc2lem 10485 axcclem 10494 axpowndlem3 10636 isstruct2 17182 cat1 18150 plusffval 18671 grpinvfval 19008 grpsubfval 19013 mulgfval 19099 0symgefmndeq 19425 staffval 20858 scaffval 20894 lpival 21351 ipffval 21683 refun0 23538 filconn 23906 alexsubALTlem2 24071 nmfval 24616 tcphex 25264 tchnmfval 25275 legval 28606 locfinref 33801 oms0 34278 bnj105 34716 ssoninhaus 36430 onint1 36431 bj-tagex 36969 bj-1uplex 36990 rrnval 37813 lsatset 38971 mnuprdlem2 44268 mnuprdlem3 44269 dvnprodlem3 45903 ioorrnopn 46260 ioorrnopnxr 46262 ismeannd 46422 |
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