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Mirrors > Home > ILE Home > Th. List > ltletrd | Unicode version |
Description: Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.) |
Ref | Expression |
---|---|
ltadd2d.1 |
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ltadd2d.2 |
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ltadd2d.3 |
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ltletrd.4 |
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ltletrd.5 |
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Ref | Expression |
---|---|
ltletrd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltletrd.4 |
. 2
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2 | ltletrd.5 |
. 2
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3 | ltadd2d.1 |
. . 3
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4 | ltadd2d.2 |
. . 3
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5 | ltadd2d.3 |
. . 3
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6 | ltletr 7877 |
. . 3
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7 | 3, 4, 5, 6 | syl3anc 1217 |
. 2
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8 | 1, 2, 7 | mp2and 430 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltwlin 7757 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 |
This theorem is referenced by: lelttrdi 8212 lediv12a 8676 btwnapz 9205 rpgecl 9499 fznatpl1 9887 elfz1b 9901 exbtwnzlemstep 10056 ceiqle 10117 modqabs 10161 mulp1mod1 10169 seq3f1olemqsumk 10303 expgt1 10362 leexp2a 10377 bernneq3 10445 expnbnd 10446 nn0opthlem2d 10499 cvg1nlemres 10789 resqrexlemlo 10817 resqrexlemnmsq 10821 resqrexlemga 10827 abssubap0 10894 icodiamlt 10984 rpmaxcl 11027 reccn2ap 11114 divcnv 11298 cvgratnnlembern 11324 cvgratnnlemabsle 11328 efcllemp 11401 sin01bnd 11500 cos01bnd 11501 sin01gt0 11504 cos12dec 11510 eirraplem 11519 dvdslelemd 11577 dvdsbnd 11681 znnen 11947 cnopnap 12802 dedekindeulemlu 12807 suplociccreex 12810 dedekindicclemlu 12816 dedekindicc 12819 ivthinclemlopn 12822 limcimolemlt 12841 limccnp2lem 12853 coseq00topi 12964 cosordlem 12978 logdivlti 13010 |
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