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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 |
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topssnei.2 |
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Ref | Expression |
---|---|
topssnei |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2 986 |
. . . 4
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2 | simprl 521 |
. . . . . 6
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3 | simpl1 985 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | simprr 522 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | tpnei.1 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() | |
6 | 5 | neii1 12355 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 3, 4, 6 | syl2anc 409 |
. . . . . . 7
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8 | 5 | ntropn 12325 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 3, 7, 8 | syl2anc 409 |
. . . . . 6
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10 | 2, 9 | sseldd 3103 |
. . . . 5
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11 | 5 | neiss2 12350 |
. . . . . . . 8
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12 | 3, 4, 11 | syl2anc 409 |
. . . . . . 7
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13 | 5 | neiint 12353 |
. . . . . . 7
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14 | 3, 12, 7, 13 | syl3anc 1217 |
. . . . . 6
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15 | 4, 14 | mpbid 146 |
. . . . 5
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16 | opnneiss 12366 |
. . . . 5
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17 | 1, 10, 15, 16 | syl3anc 1217 |
. . . 4
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18 | 5 | ntrss2 12329 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 3, 7, 18 | syl2anc 409 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | simpl3 987 |
. . . . 5
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21 | 7, 20 | sseqtrd 3140 |
. . . 4
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22 | topssnei.2 |
. . . . 5
![]() ![]() ![]() ![]() ![]() | |
23 | 22 | ssnei2 12365 |
. . . 4
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24 | 1, 17, 19, 21, 23 | syl22anc 1218 |
. . 3
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25 | 24 | expr 373 |
. 2
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26 | 25 | ssrdv 3108 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-top 12204 df-ntr 12304 df-nei 12347 |
This theorem is referenced by: (None) |
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