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Mirrors > Home > ILE Home > Th. List > ltled | Unicode version |
Description: 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | |
ltd.2 | |
ltled.1 |
Ref | Expression |
---|---|
ltled |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltled.1 | . 2 | |
2 | ltd.1 | . . 3 | |
3 | ltd.2 | . . 3 | |
4 | ltle 8007 | . . 3 | |
5 | 2, 3, 4 | syl2anc 409 | . 2 |
6 | 1, 5 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2141 class class class wbr 3989 cr 7773 clt 7954 cle 7955 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltirr 7886 ax-pre-lttrn 7888 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 |
This theorem is referenced by: ltnsymd 8039 addgt0d 8440 lt2addd 8486 lt2msq1 8801 lediv12a 8810 ledivp1 8819 nn2ge 8911 fznatpl1 10032 exbtwnzlemex 10206 apbtwnz 10230 iseqf1olemkle 10440 expnbnd 10599 nn0ltexp2 10644 cvg1nlemres 10949 resqrexlemnm 10982 resqrexlemcvg 10983 resqrexlemglsq 10986 sqrtgt0 10998 leabs 11038 ltabs 11051 abslt 11052 absle 11053 maxabslemab 11170 2zsupmax 11189 2zinfmin 11206 xrmaxiflemab 11210 fsum3cvg3 11359 divcnv 11460 expcnvre 11466 absltap 11472 cvgratnnlemnexp 11487 cvgratnnlemmn 11488 cvgratnnlemfm 11492 mertenslemi1 11498 cos12dec 11730 dvdslelemd 11803 divalglemnn 11877 divalglemeuneg 11882 lcmgcdlem 12031 isprm5lem 12095 znege1 12132 sqrt2irraplemnn 12133 eulerthlemrprm 12183 eulerthlema 12184 4sqlem7 12336 ennnfonelemex 12369 strleund 12506 suplociccreex 13396 ivthinclemlm 13406 ivthinclemum 13407 ivthinclemlopn 13408 ivthinclemuopn 13410 ivthdec 13416 dveflem 13481 efltlemlt 13489 sin0pilem1 13496 sin0pilem2 13497 coseq0negpitopi 13551 tangtx 13553 cosq34lt1 13565 cos02pilt1 13566 apdifflemf 14078 |
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