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Mirrors > Home > ILE Home > Th. List > neiint | Unicode version |
Description: An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
neifval.1 |
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Ref | Expression |
---|---|
neiint |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neifval.1 |
. . . . 5
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2 | 1 | isnei 12150 |
. . . 4
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3 | 2 | 3adant3 982 |
. . 3
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4 | 3 | 3anibar 1130 |
. 2
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5 | simprrl 511 |
. . . . 5
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6 | 1 | ssntr 12128 |
. . . . . . 7
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7 | 6 | 3adantl2 1119 |
. . . . . 6
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8 | 7 | adantrrl 475 |
. . . . 5
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9 | 5, 8 | sstrd 3071 |
. . . 4
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10 | 9 | rexlimdvaa 2522 |
. . 3
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11 | simpl1 965 |
. . . . . 6
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12 | simpl3 967 |
. . . . . 6
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13 | 1 | ntropn 12123 |
. . . . . 6
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14 | 11, 12, 13 | syl2anc 406 |
. . . . 5
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15 | simpr 109 |
. . . . 5
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16 | 1 | ntrss2 12127 |
. . . . . 6
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17 | 11, 12, 16 | syl2anc 406 |
. . . . 5
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18 | sseq2 3085 |
. . . . . . 7
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19 | sseq1 3084 |
. . . . . . 7
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20 | 18, 19 | anbi12d 462 |
. . . . . 6
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21 | 20 | rspcev 2758 |
. . . . 5
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22 | 14, 15, 17, 21 | syl12anc 1195 |
. . . 4
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23 | 22 | ex 114 |
. . 3
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24 | 10, 23 | impbid 128 |
. 2
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25 | 4, 24 | bitrd 187 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-top 12002 df-ntr 12102 df-nei 12145 |
This theorem is referenced by: topssnei 12168 iscnp4 12223 |
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