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| Mirrors > Home > MPE Home > Th. List > isopn3i | Structured version Visualization version GIF version | ||
| Description: An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.) |
| Ref | Expression |
|---|---|
| isopn3i | β’ ((π½ β Top β§ π β π½) β ((intβπ½)βπ) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpr 483 | . 2 β’ ((π½ β Top β§ π β π½) β π β π½) | |
| 2 | elssuni 4940 | . . 3 β’ (π β π½ β π β βͺ π½) | |
| 3 | eqid 2730 | . . . 4 β’ βͺ π½ = βͺ π½ | |
| 4 | 3 | isopn3 22790 | . . 3 β’ ((π½ β Top β§ π β βͺ π½) β (π β π½ β ((intβπ½)βπ) = π)) |
| 5 | 2, 4 | sylan2 591 | . 2 β’ ((π½ β Top β§ π β π½) β (π β π½ β ((intβπ½)βπ) = π)) |
| 6 | 1, 5 | mpbid 231 | 1 β’ ((π½ β Top β§ π β π½) β ((intβπ½)βπ) = π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 β wss 3947 βͺ cuni 4907 βcfv 6542 Topctop 22615 intcnt 22741 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-top 22616 df-ntr 22744 |
| This theorem is referenced by: maxlp 22871 cnntr 22999 bcth2 25078 dvrec 25707 dvmptres 25715 dvcnvlem 25728 dvlip 25745 dvlipcn 25746 dvlip2 25747 dvne0 25763 lhop2 25767 lhop 25768 psercn 26174 dvlog 26395 dvlog2 26397 cxpcn3 26492 efrlim 26710 lgamgulmlem2 26770 cvmlift2lem11 34602 cvmlift2lem12 34603 dvrelog3 41236 binomcxplemdvbinom 43414 binomcxplemnotnn0 43417 limciccioolb 44635 limcicciooub 44651 limcresiooub 44656 limcresioolb 44657 dirkercncflem2 45118 fourierdlem32 45153 fourierdlem33 45154 fourierdlem48 45168 fourierdlem49 45169 fourierdlem62 45182 fouriersw 45245 |
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