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Mirrors > Home > ILE Home > Th. List > ltletr | Unicode version |
Description: Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
ltletr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 522 | . . . 4 | |
2 | simpl2 990 | . . . . 5 | |
3 | simpl3 991 | . . . . 5 | |
4 | lenlt 7965 | . . . . 5 | |
5 | 2, 3, 4 | syl2anc 409 | . . . 4 |
6 | 1, 5 | mpbid 146 | . . 3 |
7 | simprl 521 | . . . 4 | |
8 | axltwlin 7957 | . . . . 5 | |
9 | 8 | adantr 274 | . . . 4 |
10 | 7, 9 | mpd 13 | . . 3 |
11 | 6, 10 | ecased 1338 | . 2 |
12 | 11 | ex 114 | 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 cr 7743 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-ltwlin 7857 |
This theorem depends on definitions: df-bi 116 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-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: ltletri 7996 ltletrd 8312 ltleadd 8335 nngt0 8873 nnrecgt0 8886 elnnnn0c 9150 elnnz1 9205 zltp1le 9236 uz3m2nn 9502 ledivge1le 9653 addlelt 9695 zltaddlt1le 9934 elfz1b 10015 elfzodifsumelfzo 10126 ssfzo12bi 10150 cos01gt0 11689 oddge22np1 11803 nn0seqcvgd 11952 coprm 12053 logdivlti 13343 |
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