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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 4662 | . 2 | |
2 | df-met 12147 | . . 3 | |
3 | 2 | releqi 4617 | . 2 |
4 | 1, 3 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wral 2414 crab 2418 cvv 2681 class class class wbr 3924 cmpt 3984 cxp 4532 wrel 4539 (class class class)co 5767 cmap 6535 cr 7612 cc0 7613 caddc 7616 cle 7794 cmet 12139 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-mpt 3986 df-xp 4540 df-rel 4541 df-met 12147 |
This theorem is referenced by: metflem 12507 ismet2 12512 |
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