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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 14104 |
. . . 4
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3 | 2 | 3adant3 1019 |
. . 3
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4 | 3 | 3anibar 1167 |
. 2
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5 | simprrl 539 |
. . . . 5
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6 | 1 | ssntr 14082 |
. . . . . . 7
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7 | 6 | 3adantl2 1156 |
. . . . . 6
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8 | 7 | adantrrl 486 |
. . . . 5
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9 | 5, 8 | sstrd 3180 |
. . . 4
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10 | 9 | rexlimdvaa 2608 |
. . 3
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11 | simpl1 1002 |
. . . . . 6
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12 | simpl3 1004 |
. . . . . 6
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13 | 1 | ntropn 14077 |
. . . . . 6
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14 | 11, 12, 13 | syl2anc 411 |
. . . . 5
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15 | simpr 110 |
. . . . 5
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16 | 1 | ntrss2 14081 |
. . . . . 6
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17 | 11, 12, 16 | syl2anc 411 |
. . . . 5
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18 | sseq2 3194 |
. . . . . . 7
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19 | sseq1 3193 |
. . . . . . 7
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20 | 18, 19 | anbi12d 473 |
. . . . . 6
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21 | 20 | rspcev 2856 |
. . . . 5
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22 | 14, 15, 17, 21 | syl12anc 1247 |
. . . 4
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23 | 22 | ex 115 |
. . 3
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24 | 10, 23 | impbid 129 |
. 2
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25 | 4, 24 | bitrd 188 |
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 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-top 13958 df-ntr 14056 df-nei 14099 |
This theorem is referenced by: topssnei 14122 iscnp4 14178 |
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