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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 |
Ref | Expression |
---|---|
ntrin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 3296 | . . . . 5 | |
2 | clscld.1 | . . . . . 6 | |
3 | 2 | ntrss 12288 | . . . . 5 |
4 | 1, 3 | mp3an3 1304 | . . . 4 |
5 | 4 | 3adant3 1001 | . . 3 |
6 | inss2 3297 | . . . . 5 | |
7 | 2 | ntrss 12288 | . . . . 5 |
8 | 6, 7 | mp3an3 1304 | . . . 4 |
9 | 8 | 3adant2 1000 | . . 3 |
10 | 5, 9 | ssind 3300 | . 2 |
11 | simp1 981 | . . 3 | |
12 | ssinss1 3305 | . . . 4 | |
13 | 12 | 3ad2ant2 1003 | . . 3 |
14 | 2 | ntropn 12286 | . . . . 5 |
15 | 14 | 3adant3 1001 | . . . 4 |
16 | 2 | ntropn 12286 | . . . . 5 |
17 | 16 | 3adant2 1000 | . . . 4 |
18 | inopn 12170 | . . . 4 | |
19 | 11, 15, 17, 18 | syl3anc 1216 | . . 3 |
20 | inss1 3296 | . . . . 5 | |
21 | 2 | ntrss2 12290 | . . . . . 6 |
22 | 21 | 3adant3 1001 | . . . . 5 |
23 | 20, 22 | sstrid 3108 | . . . 4 |
24 | inss2 3297 | . . . . 5 | |
25 | 2 | ntrss2 12290 | . . . . . 6 |
26 | 25 | 3adant2 1000 | . . . . 5 |
27 | 24, 26 | sstrid 3108 | . . . 4 |
28 | 23, 27 | ssind 3300 | . . 3 |
29 | 2 | ssntr 12291 | . . 3 |
30 | 11, 13, 19, 28, 29 | syl22anc 1217 | . 2 |
31 | 10, 30 | eqssd 3114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 962 wceq 1331 wcel 1480 cin 3070 wss 3071 cuni 3736 cfv 5123 ctop 12164 cnt 12262 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-top 12165 df-ntr 12265 |
This theorem is referenced by: (None) |
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