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Mirrors > Home > ILE Home > Th. List > xrlenlt | Unicode version |
Description: 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
xrlenlt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3930 | . . 3 | |
2 | opelxpi 4571 | . . . 4 | |
3 | df-le 7809 | . . . . . . 7 | |
4 | 3 | eleq2i 2206 | . . . . . 6 |
5 | eldif 3080 | . . . . . 6 | |
6 | 4, 5 | bitri 183 | . . . . 5 |
7 | 6 | baib 904 | . . . 4 |
8 | 2, 7 | syl 14 | . . 3 |
9 | 1, 8 | syl5bb 191 | . 2 |
10 | opelcnvg 4719 | . . . 4 | |
11 | df-br 3930 | . . . 4 | |
12 | 10, 11 | syl6rbbr 198 | . . 3 |
13 | 12 | notbid 656 | . 2 |
14 | 9, 13 | bitr4d 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 1480 cdif 3068 cop 3530 class class class wbr 3929 cxp 4537 ccnv 4538 cxr 7802 clt 7803 cle 7804 |
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 603 ax-in2 604 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-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-le 7809 |
This theorem is referenced by: lenlt 7843 pnfge 9578 mnfle 9581 xrltle 9587 xrleid 9589 xrletri3 9591 xrlelttr 9592 xrltletr 9593 xrletr 9594 xgepnf 9602 xleneg 9623 xltadd1 9662 xsubge0 9667 xleaddadd 9673 iccid 9711 icc0r 9712 icodisj 9778 ioodisj 9779 ioo0 10040 ico0 10042 ioc0 10043 leisorel 10583 xrmaxleim 11016 xrmaxiflemval 11022 xrmaxlesup 11031 xrmaxaddlem 11032 xrminmax 11037 bldisj 12573 bdxmet 12673 bdbl 12675 |
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