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| Mirrors > Home > MPE Home > Th. List > 0le2 | Structured version Visualization version GIF version | ||
| Description: The number 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 11786 | . . 3 ⊢ 0 ≤ 1 | |
| 2 | 1re 11261 | . . . 4 ⊢ 1 ∈ ℝ | |
| 3 | 2, 2 | addge0i 11803 | . . 3 ⊢ ((0 ≤ 1 ∧ 0 ≤ 1) → 0 ≤ (1 + 1)) |
| 4 | 1, 1, 3 | mp2an 692 | . 2 ⊢ 0 ≤ (1 + 1) |
| 5 | df-2 12329 | . 2 ⊢ 2 = (1 + 1) | |
| 6 | 4, 5 | breqtrri 5170 | 1 ⊢ 0 ≤ 2 |
| Colors of variables: wff setvar class |
| Syntax hints: class class class wbr 5143 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ≤ cle 11296 2c2 12321 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-2 12329 |
| This theorem is referenced by: expubnd 14217 4bc2eq6 14368 sqrt4 15311 sqrt2gt1lt2 15313 sqreulem 15398 amgm2 15408 efcllem 16113 ege2le3 16126 cos2bnd 16224 evennn2n 16388 6gcd4e2 16575 isprm7 16745 efgredleme 19761 abvtrivd 20833 zringndrg 21479 iihalf1 24958 minveclem2 25460 sincos4thpi 26555 tan4thpiOLD 26557 2irrexpq 26773 log2tlbnd 26988 ppisval 27147 bposlem1 27328 bposlem8 27335 bposlem9 27336 lgslem1 27341 m1lgs 27432 2lgslem1a1 27433 2lgslem4 27450 2sqlem11 27473 2sq2 27477 2sqreultlem 27491 2sqreunnltlem 27494 dchrisumlem3 27535 mulog2sumlem2 27579 log2sumbnd 27588 chpdifbndlem1 27597 usgr2pthlem 29783 pthdlem2 29788 ex-abs 30474 nrt2irr 30492 ipidsq 30729 minvecolem2 30894 normpar2i 31175 nexple 32833 wrdt2ind 32938 sqsscirc1 33907 eulerpartlemgc 34364 knoppndvlem10 36522 knoppndvlem11 36523 knoppndvlem14 36526 lcm2un 42015 aks4d1p1p7 42075 posbezout 42101 2ap1caineq 42146 pellexlem2 42841 sqrtcval 43654 imo72b2lem0 44178 sumnnodd 45645 0ellimcdiv 45664 stoweidlem26 46041 wallispilem4 46083 wallispi 46085 wallispi2lem1 46086 wallispi2 46088 stirlinglem1 46089 stirlinglem5 46093 stirlinglem6 46094 stirlinglem7 46095 stirlinglem11 46099 stirlinglem15 46103 fourierdlem68 46189 fouriersw 46246 smfmullem4 46809 lighneallem4a 47595 fpprel2 47728 |
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