Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xmetrel | Unicode version |
Description: The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.) |
Ref | Expression |
---|---|
xmetrel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptrel 4637 | . 2 | |
2 | df-xmet 12084 | . . 3 | |
3 | 2 | releqi 4592 | . 2 |
4 | 1, 3 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wral 2393 crab 2397 cvv 2660 class class class wbr 3899 cmpt 3959 cxp 4507 wrel 4514 (class class class)co 5742 cmap 6510 cc0 7588 cxr 7767 cle 7769 cxad 9525 cxmet 12076 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-mpt 3961 df-xp 4515 df-rel 4516 df-xmet 12084 |
This theorem is referenced by: ismet2 12450 xmeteq0 12455 xmettri2 12457 xmetpsmet 12465 xmetres2 12475 blex 12483 blval 12485 blf 12506 mopnval 12538 comet 12595 |
Copyright terms: Public domain | W3C validator |