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Theorem ltle 7105

Description: 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)

**Assertion**
Ref	Expression
ltle	$/\$

**Proof of Theorem ltle**
Step	Hyp	Ref	Expression
1		ltnsym 7104	. 2 $/\$
2		lenlt 7094	. 2 $/\$
3	1, 2	sylibrd 158	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97

wcel 1393 class class class wbr 3764

cr 6888

clt 7060

cle 7061

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 ax-pre-lttrn 6998

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066

This theorem is referenced by: ltlei 7119 ltled 7135 ltleap 7621 lep1 7811 lem1 7813 letrp1 7814 ltmul12a 7826 bndndx 8180 nn0ge0 8207 zletric 8289 zlelttric 8290 zltnle 8291 zleloe 8292 zdcle 8317 uzind 8349 fnn0ind 8354 eluz2b2 8540 rpge0 8595 difelfznle 8993 elfzouz2 9017 elfzo0le 9041 fzosplitprm1 9090 fzostep1 9093 qletric 9099 qlelttric 9100 qltnle 9101 expgt1 9293 expnlbnd2 9374 caucvgrelemcau 9579 resqrexlemdecn 9610 mulcn2 9833