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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 12435 | . 2 ⊢ 1 < 2 | |
2 | 2lt3 12436 | . 2 ⊢ 2 < 3 | |
3 | 1re 11264 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 12338 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 3re 12344 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 3, 4, 5 | lttri 11390 | . 2 ⊢ ((1 < 2 ∧ 2 < 3) → 1 < 3) |
7 | 1, 2, 6 | mp2an 690 | 1 ⊢ 1 < 3 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5153 1c1 11159 < clt 11298 2c2 12319 3c3 12320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-2 12327 df-3 12328 |
This theorem is referenced by: 1le3 12476 fztpval 13617 expnass 14226 s4fv1 14905 f1oun2prg 14926 sin01gt0 16192 rpnnen2lem3 16218 rpnnen2lem9 16224 3prm 16695 6nprm 17112 7prm 17113 9nprm 17115 13prm 17118 19prm 17120 prmlem2 17122 37prm 17123 43prm 17124 139prm 17126 163prm 17127 631prm 17129 basendxnmulrndx 17309 basendxnmulrndxOLD 17310 opprbasOLD 20324 log2cnv 26972 cxploglim2 27007 2lgslem3 27433 dchrvmasumlem2 27527 pntibndlem1 27618 tgcgr4 28458 axlowdimlem16 28891 usgrexmpldifpr 29194 upgr3v3e3cycl 30113 upgr4cycl4dv4e 30118 konigsberglem2 30186 konigsberglem3 30187 konigsberglem5 30189 frgrogt3nreg 30330 ex-dif 30356 ex-pss 30361 ex-res 30374 evl1deg3 33450 2sqr3minply 33607 aks4d1p1p3 41768 aks4d1p1p2 41769 aks4d1p1p4 41770 aks4d1p3 41777 acos1half 42328 rabren3dioph 42472 jm2.23 42654 mnringbasedOLD 43886 stoweidlem34 45655 stoweidlem42 45663 smfmullem4 46415 fmtno4prmfac193 47145 3ndvds4 47167 127prm 47171 nnsum4primesodd 47368 nnsum4primesoddALTV 47369 sepfsepc 48261 |
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