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Mirrors > Home > ILE Home > Th. List > topssnei | Unicode version |
Description: A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
tpnei.1 | |
topssnei.2 |
Ref | Expression |
---|---|
topssnei |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2 985 | . . . 4 | |
2 | simprl 520 | . . . . . 6 | |
3 | simpl1 984 | . . . . . . 7 | |
4 | simprr 521 | . . . . . . . 8 | |
5 | tpnei.1 | . . . . . . . . 9 | |
6 | 5 | neii1 12316 | . . . . . . . 8 |
7 | 3, 4, 6 | syl2anc 408 | . . . . . . 7 |
8 | 5 | ntropn 12286 | . . . . . . 7 |
9 | 3, 7, 8 | syl2anc 408 | . . . . . 6 |
10 | 2, 9 | sseldd 3098 | . . . . 5 |
11 | 5 | neiss2 12311 | . . . . . . . 8 |
12 | 3, 4, 11 | syl2anc 408 | . . . . . . 7 |
13 | 5 | neiint 12314 | . . . . . . 7 |
14 | 3, 12, 7, 13 | syl3anc 1216 | . . . . . 6 |
15 | 4, 14 | mpbid 146 | . . . . 5 |
16 | opnneiss 12327 | . . . . 5 | |
17 | 1, 10, 15, 16 | syl3anc 1216 | . . . 4 |
18 | 5 | ntrss2 12290 | . . . . 5 |
19 | 3, 7, 18 | syl2anc 408 | . . . 4 |
20 | simpl3 986 | . . . . 5 | |
21 | 7, 20 | sseqtrd 3135 | . . . 4 |
22 | topssnei.2 | . . . . 5 | |
23 | 22 | ssnei2 12326 | . . . 4 |
24 | 1, 17, 19, 21, 23 | syl22anc 1217 | . . 3 |
25 | 24 | expr 372 | . 2 |
26 | 25 | ssrdv 3103 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wss 3071 cuni 3736 cfv 5123 ctop 12164 cnt 12262 cnei 12307 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-top 12165 df-ntr 12265 df-nei 12308 |
This theorem is referenced by: (None) |
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