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| Mirrors > Home > MPE Home > Th. List > ntropn | Structured version Visualization version GIF version | ||
| Description: The interior of a subset of a topology's underlying set is open. (Contributed by NM, 11-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
| Ref | Expression |
|---|---|
| clscld.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| ntropn | β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) β π½) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clscld.1 | . . 3 β’ π = βͺ π½ | |
| 2 | 1 | ntrval 22540 | . 2 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| 3 | inss1 4229 | . . . 4 β’ (π½ β© π« π) β π½ | |
| 4 | uniopn 22399 | . . . 4 β’ ((π½ β Top β§ (π½ β© π« π) β π½) β βͺ (π½ β© π« π) β π½) | |
| 5 | 3, 4 | mpan2 690 | . . 3 β’ (π½ β Top β βͺ (π½ β© π« π) β π½) |
| 6 | 5 | adantr 482 | . 2 β’ ((π½ β Top β§ π β π) β βͺ (π½ β© π« π) β π½) |
| 7 | 2, 6 | eqeltrd 2834 | 1 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) β π½) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β© cin 3948 β wss 3949 π« cpw 4603 βͺ cuni 4909 βcfv 6544 Topctop 22395 intcnt 22521 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-top 22396 df-ntr 22524 |
| This theorem is referenced by: ntrval2 22555 ntrss3 22564 ntrin 22565 cmclsopn 22566 cmntrcld 22567 isopn3 22570 ntridm 22572 neiint 22608 topssnei 22628 maxlp 22651 restntr 22686 iscnp4 22767 cnntri 22775 cnprest 22793 llycmpkgen2 23054 xkococnlem 23163 flimopn 23479 fclsneii 23521 fcfnei 23539 subgntr 23611 iccntr 24337 rectbntr0 24348 bcthlem5 24845 bcth3 24848 limcflf 25398 perfdvf 25420 ubthlem1 30123 cvmlift2lem12 34305 opnregcld 35215 ntrrn 42873 toplatglb 47626 |
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