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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 7638 | . 2 ⊢ 0 ∈ ℝ | |
2 | 1re 7637 | . 2 ⊢ 1 ∈ ℝ | |
3 | 0lt1 7760 | . 2 ⊢ 0 < 1 | |
4 | 1, 2, 3 | ltleii 7737 | 1 ⊢ 0 ≤ 1 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3875 0cc0 7500 1c1 7501 ≤ cle 7673 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1re 7589 ax-addrcl 7592 ax-0lt1 7601 ax-rnegex 7604 ax-pre-ltirr 7607 ax-pre-lttrn 7609 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-xp 4483 df-cnv 4485 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 |
This theorem is referenced by: lemulge11 8482 sup3exmid 8573 0le2 8668 1eluzge0 9219 0elunit 9610 1elunit 9611 fldiv4p1lem1div2 9919 q1mod 9970 expge0 10170 expge1 10171 faclbnd3 10330 sqrt1 10658 sqrt2gt1lt2 10661 abs1 10684 cvgratnnlembern 11131 ege2le3 11175 sinbnd 11257 cosbnd 11258 cos2bnd 11265 nn0oddm1d2 11401 flodddiv4 11426 sqnprm 11609 sqrt2irrap 11650 nn0sqrtelqelz 11676 trilpolemclim 12813 trilpolemlt1 12818 |
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