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Mirrors > Home > MPE Home > Th. List > 1lt3 | Structured version Visualization version GIF version |
Description: 1 is less than 3. (Contributed by NM, 26-Sep-2010.) |
Ref | Expression |
---|---|
1lt3 | ⊢ 1 < 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2 11557 | . 2 ⊢ 1 < 2 | |
2 | 2lt3 11558 | . 2 ⊢ 2 < 3 | |
3 | 1re 10378 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 11453 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 3re 11459 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10504 | . 2 ⊢ ((1 < 2 ∧ 2 < 3) → 1 < 3) |
7 | 1, 2, 6 | mp2an 682 | 1 ⊢ 1 < 3 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4888 1c1 10275 < clt 10413 2c2 11434 3c3 11435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-2 11442 df-3 11443 |
This theorem is referenced by: 1le3 11598 fztpval 12724 expnass 13293 s4fv1 14051 f1oun2prg 14072 sin01gt0 15326 rpnnen2lem3 15353 rpnnen2lem9 15359 3prm 15815 6nprm 16219 7prm 16220 9nprm 16222 13prm 16225 19prm 16227 prmlem2 16229 37prm 16230 43prm 16231 139prm 16233 163prm 16234 631prm 16236 basendxnmulrndx 16395 ressmulr 16402 opprbas 19020 matbas 20627 log2cnv 25127 cxploglim2 25161 2lgslem3 25585 dchrvmasumlem2 25643 pntibndlem1 25734 tgcgr4 25886 axlowdimlem16 26310 usgrexmpldifpr 26609 upgr3v3e3cycl 27587 upgr4cycl4dv4e 27592 konigsberglem2 27663 konigsberglem3 27664 konigsberglem5 27666 frgrogt3nreg 27833 ex-dif 27859 ex-pss 27864 ex-res 27877 rabren3dioph 38349 jm2.23 38532 stoweidlem34 41188 stoweidlem42 41196 smfmullem4 41938 fmtno4prmfac193 42516 3ndvds4 42541 127prm 42546 nnsum4primesodd 42719 nnsum4primesoddALTV 42720 |
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