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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 970 | . . . 4 | |
2 | simprl 505 | . . . . . 6 | |
3 | simpl1 969 | . . . . . . 7 | |
4 | simprr 506 | . . . . . . . 8 | |
5 | tpnei.1 | . . . . . . . . 9 | |
6 | 5 | neii1 12243 | . . . . . . . 8 |
7 | 3, 4, 6 | syl2anc 408 | . . . . . . 7 |
8 | 5 | ntropn 12213 | . . . . . . 7 |
9 | 3, 7, 8 | syl2anc 408 | . . . . . 6 |
10 | 2, 9 | sseldd 3068 | . . . . 5 |
11 | 5 | neiss2 12238 | . . . . . . . 8 |
12 | 3, 4, 11 | syl2anc 408 | . . . . . . 7 |
13 | 5 | neiint 12241 | . . . . . . 7 |
14 | 3, 12, 7, 13 | syl3anc 1201 | . . . . . 6 |
15 | 4, 14 | mpbid 146 | . . . . 5 |
16 | opnneiss 12254 | . . . . 5 | |
17 | 1, 10, 15, 16 | syl3anc 1201 | . . . 4 |
18 | 5 | ntrss2 12217 | . . . . 5 |
19 | 3, 7, 18 | syl2anc 408 | . . . 4 |
20 | simpl3 971 | . . . . 5 | |
21 | 7, 20 | sseqtrd 3105 | . . . 4 |
22 | topssnei.2 | . . . . 5 | |
23 | 22 | ssnei2 12253 | . . . 4 |
24 | 1, 17, 19, 21, 23 | syl22anc 1202 | . . 3 |
25 | 24 | expr 372 | . 2 |
26 | 25 | ssrdv 3073 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wss 3041 cuni 3706 cfv 5093 ctop 12091 cnt 12189 cnei 12234 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-top 12092 df-ntr 12192 df-nei 12235 |
This theorem is referenced by: (None) |
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