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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 3355 |
. . . . 5
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2 | clscld.1 |
. . . . . 6
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3 | 2 | ntrss 13283 |
. . . . 5
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4 | 1, 3 | mp3an3 1326 |
. . . 4
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5 | 4 | 3adant3 1017 |
. . 3
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6 | inss2 3356 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 2 | ntrss 13283 |
. . . . 5
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8 | 6, 7 | mp3an3 1326 |
. . . 4
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9 | 8 | 3adant2 1016 |
. . 3
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10 | 5, 9 | ssind 3359 |
. 2
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11 | simp1 997 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | ssinss1 3364 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 12 | 3ad2ant2 1019 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 2 | ntropn 13281 |
. . . . 5
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15 | 14 | 3adant3 1017 |
. . . 4
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16 | 2 | ntropn 13281 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | 3adant2 1016 |
. . . 4
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18 | inopn 13165 |
. . . 4
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19 | 11, 15, 17, 18 | syl3anc 1238 |
. . 3
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20 | inss1 3355 |
. . . . 5
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21 | 2 | ntrss2 13285 |
. . . . . 6
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22 | 21 | 3adant3 1017 |
. . . . 5
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23 | 20, 22 | sstrid 3166 |
. . . 4
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24 | inss2 3356 |
. . . . 5
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25 | 2 | ntrss2 13285 |
. . . . . 6
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26 | 25 | 3adant2 1016 |
. . . . 5
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27 | 24, 26 | sstrid 3166 |
. . . 4
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28 | 23, 27 | ssind 3359 |
. . 3
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29 | 2 | ssntr 13286 |
. . 3
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30 | 11, 13, 19, 28, 29 | syl22anc 1239 |
. 2
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31 | 10, 30 | eqssd 3172 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-top 13160 df-ntr 13260 |
This theorem is referenced by: (None) |
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