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Mirrors > Home > ILE Home > Th. List > ntrin | Unicode version |
Description: A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.) |
Ref | Expression |
---|---|
clscld.1 |
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Ref | Expression |
---|---|
ntrin |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 3301 |
. . . . 5
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2 | clscld.1 |
. . . . . 6
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3 | 2 | ntrss 12327 |
. . . . 5
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4 | 1, 3 | mp3an3 1305 |
. . . 4
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5 | 4 | 3adant3 1002 |
. . 3
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6 | inss2 3302 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 2 | ntrss 12327 |
. . . . 5
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8 | 6, 7 | mp3an3 1305 |
. . . 4
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9 | 8 | 3adant2 1001 |
. . 3
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10 | 5, 9 | ssind 3305 |
. 2
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11 | simp1 982 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | ssinss1 3310 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 12 | 3ad2ant2 1004 |
. . 3
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14 | 2 | ntropn 12325 |
. . . . 5
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15 | 14 | 3adant3 1002 |
. . . 4
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16 | 2 | ntropn 12325 |
. . . . 5
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17 | 16 | 3adant2 1001 |
. . . 4
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18 | inopn 12209 |
. . . 4
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19 | 11, 15, 17, 18 | syl3anc 1217 |
. . 3
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20 | inss1 3301 |
. . . . 5
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21 | 2 | ntrss2 12329 |
. . . . . 6
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22 | 21 | 3adant3 1002 |
. . . . 5
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23 | 20, 22 | sstrid 3113 |
. . . 4
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24 | inss2 3302 |
. . . . 5
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25 | 2 | ntrss2 12329 |
. . . . . 6
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26 | 25 | 3adant2 1001 |
. . . . 5
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27 | 24, 26 | sstrid 3113 |
. . . 4
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28 | 23, 27 | ssind 3305 |
. . 3
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29 | 2 | ssntr 12330 |
. . 3
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30 | 11, 13, 19, 28, 29 | syl22anc 1218 |
. 2
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31 | 10, 30 | eqssd 3119 |
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 |
This theorem is referenced by: (None) |
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