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Mirrors > Home > ILE Home > Th. List > 0le1 | GIF version |
Description: 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
0le1 | ⊢ 0 ≤ 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7988 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 7987 | . 2 ⊢ 1 ∈ ℝ | |
3 | 0lt1 8115 | . 2 ⊢ 0 < 1 | |
4 | 1, 2, 3 | ltleii 8091 | 1 ⊢ 0 ≤ 1 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4018 0cc0 7842 1c1 7843 ≤ cle 8024 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1re 7936 ax-addrcl 7939 ax-0lt1 7948 ax-rnegex 7951 ax-pre-ltirr 7954 ax-pre-lttrn 7956 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 |
This theorem is referenced by: lemulge11 8854 sup3exmid 8945 0le2 9040 1eluzge0 9606 0elunit 10018 1elunit 10019 fldiv4p1lem1div2 10338 q1mod 10389 expge0 10590 expge1 10591 faclbnd3 10758 sqrt1 11090 sqrt2gt1lt2 11093 abs1 11116 cvgratnnlembern 11566 fprodge0 11680 fprodge1 11682 ege2le3 11714 sinbnd 11795 cosbnd 11796 cos2bnd 11803 nn0oddm1d2 11949 flodddiv4 11974 sqnprm 12171 isprm5lem 12176 sqrt2irrap 12215 nn0sqrtelqelz 12241 pythagtriplem3 12302 sinhalfpilem 14689 zabsle1 14878 lgslem2 14880 lgsfcl2 14885 lgsdir2lem1 14907 lgsne0 14917 lgsdinn0 14927 m1lgs 14930 trilpolemclim 15263 trilpolemlt1 15268 nconstwlpolemgt0 15291 |
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