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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 11151 | . . 3 ⊢ 0 ≤ 1 | |
2 | 1re 10629 | . . . 4 ⊢ 1 ∈ ℝ | |
3 | 2, 2 | addge0i 11168 | . . 3 ⊢ ((0 ≤ 1 ∧ 0 ≤ 1) → 0 ≤ (1 + 1)) |
4 | 1, 1, 3 | mp2an 688 | . 2 ⊢ 0 ≤ (1 + 1) |
5 | df-2 11688 | . 2 ⊢ 2 = (1 + 1) | |
6 | 4, 5 | breqtrri 5084 | 1 ⊢ 0 ≤ 2 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5057 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 ≤ cle 10664 2c2 11680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-2 11688 |
This theorem is referenced by: expubnd 13529 4bc2eq6 13677 sqrt4 14620 sqrt2gt1lt2 14622 sqreulem 14707 amgm2 14717 efcllem 15419 ege2le3 15431 cos2bnd 15529 evennn2n 15688 6gcd4e2 15874 isprm7 16040 efgredleme 18798 abvtrivd 19540 zringndrg 20565 iihalf1 23462 minveclem2 23956 sincos4thpi 25026 tan4thpi 25027 2irrexpq 25240 log2tlbnd 25450 ppisval 25608 bposlem1 25787 bposlem8 25794 bposlem9 25795 lgslem1 25800 m1lgs 25891 2lgslem1a1 25892 2lgslem4 25909 2sqlem11 25932 2sq2 25936 2sqreultlem 25950 2sqreunnltlem 25953 dchrisumlem3 25994 mulog2sumlem2 26038 log2sumbnd 26047 chpdifbndlem1 26056 usgr2pthlem 27471 pthdlem2 27476 ex-abs 28161 ipidsq 28414 minvecolem2 28579 normpar2i 28860 wrdt2ind 30554 sqsscirc1 31050 nexple 31167 eulerpartlemgc 31519 knoppndvlem10 33757 knoppndvlem11 33758 knoppndvlem14 33761 pellexlem2 39305 imo72b2lem0 40394 sumnnodd 41787 0ellimcdiv 41806 stoweidlem26 42188 wallispilem4 42230 wallispi 42232 wallispi2lem1 42233 wallispi2 42235 stirlinglem1 42236 stirlinglem5 42240 stirlinglem6 42241 stirlinglem7 42242 stirlinglem11 42246 stirlinglem15 42250 fourierdlem68 42336 fouriersw 42393 smfmullem4 42946 lighneallem4a 43650 fpprel2 43783 |
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