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Mirrors > Home > ILE Home > Th. List > elblps | Unicode version |
Description: Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
Ref | Expression |
---|---|
elblps | PsMet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blvalps 12788 | . . 3 PsMet | |
2 | 1 | eleq2d 2227 | . 2 PsMet |
3 | oveq2 5832 | . . . 4 | |
4 | 3 | breq1d 3975 | . . 3 |
5 | 4 | elrab 2868 | . 2 |
6 | 2, 5 | bitrdi 195 | 1 PsMet |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 crab 2439 class class class wbr 3965 cfv 5170 (class class class)co 5824 cxr 7911 clt 7912 PsMetcpsmet 12379 cbl 12382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-map 6595 df-pnf 7914 df-mnf 7915 df-xr 7916 df-psmet 12387 df-bl 12390 |
This theorem is referenced by: elbl2ps 12792 xblpnfps 12798 xblss2ps 12804 xblcntrps 12813 blssps 12827 |
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