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Mirrors > Home > ILE Home > Th. List > opnneiss | Unicode version |
Description: An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.) |
Ref | Expression |
---|---|
opnneiss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 984 | . 2 | |
2 | eqid 2157 | . . . . 5 | |
3 | 2 | eltopss 12503 | . . . 4 |
4 | sstr 3136 | . . . . 5 | |
5 | 4 | ancoms 266 | . . . 4 |
6 | 3, 5 | stoic3 1411 | . . 3 |
7 | 2 | opnneissb 12651 | . . 3 |
8 | 6, 7 | syld3an3 1265 | . 2 |
9 | 1, 8 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 963 wcel 2128 wss 3102 cuni 3774 cfv 5173 ctop 12491 cnei 12634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-top 12492 df-nei 12635 |
This theorem is referenced by: opnneip 12655 tpnei 12656 topssnei 12658 opnneiid 12660 neissex 12661 |
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