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Mirrors > Home > ILE Home > Th. List > 0le2 | GIF version |
Description: 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) |
Ref | Expression |
---|---|
0le2 | ⊢ 0 ≤ 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0le1 8415 | . . 3 ⊢ 0 ≤ 1 | |
2 | 1re 7934 | . . . 4 ⊢ 1 ∈ ℝ | |
3 | 2, 2 | addge0i 8423 | . . 3 ⊢ ((0 ≤ 1 ∧ 0 ≤ 1) → 0 ≤ (1 + 1)) |
4 | 1, 1, 3 | mp2an 426 | . 2 ⊢ 0 ≤ (1 + 1) |
5 | df-2 8954 | . 2 ⊢ 2 = (1 + 1) | |
6 | 4, 5 | breqtrri 4027 | 1 ⊢ 0 ≤ 2 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4000 (class class class)co 5868 0cc0 7789 1c1 7790 + caddc 7792 ≤ cle 7970 2c2 8946 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4628 df-cnv 4630 df-iota 5173 df-fv 5219 df-ov 5871 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-2 8954 |
This theorem is referenced by: expubnd 10550 4bc2eq6 10725 sqrt4 11027 sqrt2gt1lt2 11029 amgm2 11098 bdtrilem 11218 ege2le3 11650 cos2bnd 11739 evennn2n 11858 6gcd4e2 11966 sqrt2irrlem 12131 sqrt2irraplemnn 12149 oddennn 12363 sincos4thpi 13894 lgslem1 14034 |
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