Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 486 | . 2 β’ ((π½ β Top β§ π β π½) β π β π½) | |
| 2 | elssuni 4942 | . . 3 β’ (π β π½ β π β βͺ π½) | |
| 3 | eqid 2733 | . . . 4 β’ βͺ π½ = βͺ π½ | |
| 4 | 3 | isopn3 22570 | . . 3 β’ ((π½ β Top β§ π β βͺ π½) β (π β π½ β ((intβπ½)βπ) = π)) |
| 5 | 2, 4 | sylan2 594 | . 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 397 = wceq 1542 β wcel 2107 β wss 3949 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: maxlp 22651 cnntr 22779 bcth2 24847 dvrec 25472 dvmptres 25480 dvcnvlem 25493 dvlip 25510 dvlipcn 25511 dvlip2 25512 dvne0 25528 lhop2 25532 lhop 25533 psercn 25938 dvlog 26159 dvlog2 26161 cxpcn3 26256 efrlim 26474 lgamgulmlem2 26534 cvmlift2lem11 34304 cvmlift2lem12 34305 dvrelog3 40930 binomcxplemdvbinom 43112 binomcxplemnotnn0 43115 limciccioolb 44337 limcicciooub 44353 limcresiooub 44358 limcresioolb 44359 dirkercncflem2 44820 fourierdlem32 44855 fourierdlem33 44856 fourierdlem48 44870 fourierdlem49 44871 fourierdlem62 44884 fouriersw 44947 |
| Copyright terms: Public domain | W3C validator |