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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 3786 | . . 3 | |
2 | opelxpi 4394 | . . . 4 | |
3 | df-le 7159 | . . . . . . 7 | |
4 | 3 | eleq2i 2145 | . . . . . 6 |
5 | eldif 2982 | . . . . . 6 | |
6 | 4, 5 | bitri 182 | . . . . 5 |
7 | 6 | baib 861 | . . . 4 |
8 | 2, 7 | syl 14 | . . 3 |
9 | 1, 8 | syl5bb 190 | . 2 |
10 | opelcnvg 4533 | . . . 4 | |
11 | df-br 3786 | . . . 4 | |
12 | 10, 11 | syl6rbbr 197 | . . 3 |
13 | 12 | notbid 624 | . 2 |
14 | 9, 13 | bitr4d 189 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wcel 1433 cdif 2970 cop 3401 class class class wbr 3785 cxp 4361 ccnv 4362 cxr 7152 clt 7153 cle 7154 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-le 7159 |
This theorem is referenced by: lenlt 7187 pnfge 8864 mnfle 8867 xrltle 8873 xrleid 8874 xrletri3 8875 xrlelttr 8876 xrltletr 8877 xrletr 8878 xleneg 8904 iccid 8948 icc0r 8949 icodisj 9014 ioodisj 9015 ioo0 9268 ico0 9270 ioc0 9271 |
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