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Mirrors > Home > ILE Home > Th. List > ntrtop | GIF version |
Description: The interior of a topology's underlying set is the entire set. (Contributed by NM, 12-Sep-2006.) |
Ref | Expression |
---|---|
clscld.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
ntrtop | β’ (π½ β Top β ((intβπ½)βπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clscld.1 | . . 3 β’ π = βͺ π½ | |
2 | 1 | topopn 13804 | . 2 β’ (π½ β Top β π β π½) |
3 | ssid 3187 | . . 3 β’ π β π | |
4 | 1 | isopn3 13921 | . . 3 β’ ((π½ β Top β§ π β π) β (π β π½ β ((intβπ½)βπ) = π)) |
5 | 3, 4 | mpan2 425 | . 2 β’ (π½ β Top β (π β π½ β ((intβπ½)βπ) = π)) |
6 | 2, 5 | mpbid 147 | 1 β’ (π½ β Top β ((intβπ½)βπ) = π) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β wb 105 = wceq 1363 β wcel 2158 β wss 3141 βͺ cuni 3821 βcfv 5228 Topctop 13793 intcnt 13889 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-top 13794 df-ntr 13892 |
This theorem is referenced by: dvidlemap 14456 dveflem 14483 |
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