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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 11232 | . 2 ⊢ 1 < 2 | |
2 | 2lt3 11233 | . 2 ⊢ 2 < 3 | |
3 | 1re 10077 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 11128 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 3re 11132 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10201 | . 2 ⊢ ((1 < 2 ∧ 2 < 3) → 1 < 3) |
7 | 1, 2, 6 | mp2an 708 | 1 ⊢ 1 < 3 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4685 1c1 9975 < clt 10112 2c2 11108 3c3 11109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-2 11117 df-3 11118 |
This theorem is referenced by: 1le3 11282 fztpval 12440 expnass 13010 s4fv1 13687 f1oun2prg 13708 sin01gt0 14964 rpnnen2lem3 14989 rpnnen2lem9 14995 3prm 15453 6nprm 15863 7prm 15864 9nprm 15866 13prm 15870 19prm 15872 prmlem2 15874 37prm 15875 43prm 15876 139prm 15878 163prm 15879 631prm 15881 basendxnmulrndx 16046 ressmulr 16053 opprbas 18675 matbas 20267 log2cnv 24716 cxploglim2 24750 2lgslem3 25174 dchrvmasumlem2 25232 pntibndlem1 25323 tgcgr4 25471 axlowdimlem16 25882 usgrexmpldifpr 26195 upgr3v3e3cycl 27158 upgr4cycl4dv4e 27163 konigsberglem2 27231 konigsberglem3 27232 konigsberglem5 27234 frgrogt3nreg 27384 ex-dif 27410 ex-pss 27415 ex-res 27428 rabren3dioph 37696 jm2.23 37880 stoweidlem34 40569 stoweidlem42 40577 smfmullem4 41322 fmtno4prmfac193 41810 3ndvds4 41835 127prm 41840 nnsum4primesodd 42009 nnsum4primesoddALTV 42010 |
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