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Mirrors > Home > ILE Home > Th. List > metrel | Unicode version |
Description: The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.) |
Ref | Expression |
---|---|
metrel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptrel 4667 | . 2 | |
2 | df-met 12163 | . . 3 | |
3 | 2 | releqi 4622 | . 2 |
4 | 1, 3 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wral 2416 crab 2420 cvv 2686 class class class wbr 3929 cmpt 3989 cxp 4537 wrel 4544 (class class class)co 5774 cmap 6542 cr 7624 cc0 7625 caddc 7628 cle 7806 cmet 12155 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-mpt 3991 df-xp 4545 df-rel 4546 df-met 12163 |
This theorem is referenced by: metflem 12523 ismet2 12528 |
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