Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > ILE Home > Th. List > xblcntr | GIF version | ||
| Description: A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
| Ref | Expression |
|---|---|
| xblcntr | β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β π β (π(ballβπ·)π )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp2 998 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β π β π) | |
| 2 | xmet0 14003 | . . . 4 β’ ((π· β (βMetβπ) β§ π β π) β (ππ·π) = 0) | |
| 3 | 2 | 3adant3 1017 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) = 0) |
| 4 | simp3r 1026 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β 0 < π ) | |
| 5 | 3, 4 | eqbrtrd 4027 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (ππ·π) < π ) |
| 6 | elbl 14031 | . . 3 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π β (π(ballβπ·)π ) β (π β π β§ (ππ·π) < π ))) | |
| 7 | 6 | 3adant3r 1235 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β (π β (π(ballβπ·)π ) β (π β π β§ (ππ·π) < π ))) |
| 8 | 1, 5, 7 | mpbir2and 944 | 1 β’ ((π· β (βMetβπ) β§ π β π β§ (π β β* β§ 0 < π )) β π β (π(ballβπ·)π )) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β wb 105 β§ w3a 978 = wceq 1353 β wcel 2148 class class class wbr 4005 βcfv 5218 (class class class)co 5878 0cc0 7814 β*cxr 7994 < clt 7995 βMetcxmet 13580 ballcbl 13582 |
| 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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-map 6653 df-pnf 7997 df-mnf 7998 df-xr 7999 df-psmet 13587 df-xmet 13588 df-bl 13590 |
| This theorem is referenced by: blcntr 14056 xblm 14057 |
| Copyright terms: Public domain | W3C validator |