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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 531 |
. . . 4
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2 | simpl2 1001 |
. . . . 5
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3 | simpl3 1002 |
. . . . 5
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4 | lenlt 8010 |
. . . . 5
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5 | 2, 3, 4 | syl2anc 411 |
. . . 4
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6 | 1, 5 | mpbid 147 |
. . 3
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7 | simprl 529 |
. . . 4
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8 | axltwlin 8002 |
. . . . 5
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9 | 8 | adantr 276 |
. . . 4
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10 | 7, 9 | mpd 13 |
. . 3
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11 | 6, 10 | ecased 1349 |
. 2
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12 | 11 | ex 115 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-pre-ltwlin 7902 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4628 df-cnv 4630 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 |
This theorem is referenced by: ltletri 8041 ltletrd 8357 ltleadd 8380 nngt0 8920 nnrecgt0 8933 elnnnn0c 9197 elnnz1 9252 zltp1le 9283 uz3m2nn 9549 ledivge1le 9700 addlelt 9742 zltaddlt1le 9981 elfz1b 10063 elfzodifsumelfzo 10174 ssfzo12bi 10198 cos01gt0 11741 oddge22np1 11856 nn0seqcvgd 12011 coprm 12114 logdivlti 13935 |
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