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Mirrors > Home > ILE Home > Th. List > xrletr | Unicode version |
Description: Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.) |
Ref | Expression |
---|---|
xrletr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 9723 | . . . . . 6 | |
2 | sowlin 4292 | . . . . . 6 | |
3 | 1, 2 | mpan 421 | . . . . 5 |
4 | 3 | 3coml 1199 | . . . 4 |
5 | orcom 718 | . . . 4 | |
6 | 4, 5 | syl6ib 160 | . . 3 |
7 | 6 | con3d 621 | . 2 |
8 | xrlenlt 7954 | . . . . 5 | |
9 | 8 | 3adant3 1006 | . . . 4 |
10 | xrlenlt 7954 | . . . . 5 | |
11 | 10 | 3adant1 1004 | . . . 4 |
12 | 9, 11 | anbi12d 465 | . . 3 |
13 | ioran 742 | . . 3 | |
14 | 12, 13 | bitr4di 197 | . 2 |
15 | xrlenlt 7954 | . . 3 | |
16 | 15 | 3adant2 1005 | . 2 |
17 | 7, 14, 16 | 3imtr4d 202 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 967 wcel 2135 class class class wbr 3976 wor 4267 cxr 7923 clt 7924 cle 7925 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-po 4268 df-iso 4269 df-xp 4604 df-cnv 4606 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 |
This theorem is referenced by: xrletrd 9739 xle2add 9806 icc0r 9853 iccss 9868 icossico 9870 iccss2 9871 iccssico 9872 bdxmet 13048 |
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