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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 991 | . . . . 5 | |
3 | simpl3 992 | . . . . 5 | |
4 | lenlt 7974 | . . . . 5 | |
5 | 2, 3, 4 | syl2anc 409 | . . . 4 |
6 | 1, 5 | mpbid 146 | . . 3 |
7 | simprl 521 | . . . 4 | |
8 | axltwlin 7966 | . . . . 5 | |
9 | 8 | adantr 274 | . . . 4 |
10 | 7, 9 | mpd 13 | . . 3 |
11 | 6, 10 | ecased 1339 | . 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 968 wcel 2136 class class class wbr 3982 cr 7752 clt 7933 cle 7934 |
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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-pre-ltwlin 7866 |
This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-xp 4610 df-cnv 4612 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 |
This theorem is referenced by: ltletri 8005 ltletrd 8321 ltleadd 8344 nngt0 8882 nnrecgt0 8895 elnnnn0c 9159 elnnz1 9214 zltp1le 9245 uz3m2nn 9511 ledivge1le 9662 addlelt 9704 zltaddlt1le 9943 elfz1b 10025 elfzodifsumelfzo 10136 ssfzo12bi 10160 cos01gt0 11703 oddge22np1 11818 nn0seqcvgd 11973 coprm 12076 logdivlti 13452 |
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