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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 12437 | . 2 ⊢ 1 < 2 | |
| 2 | 2lt3 12438 | . 2 ⊢ 2 < 3 | |
| 3 | 1re 11261 | . . 3 ⊢ 1 ∈ ℝ | |
| 4 | 2re 12340 | . . 3 ⊢ 2 ∈ ℝ | |
| 5 | 3re 12346 | . . 3 ⊢ 3 ∈ ℝ | |
| 6 | 3, 4, 5 | lttri 11387 | . 2 ⊢ ((1 < 2 ∧ 2 < 3) → 1 < 3) |
| 7 | 1, 2, 6 | mp2an 692 | 1 ⊢ 1 < 3 |
| Colors of variables: wff setvar class |
| Syntax hints: class class class wbr 5143 1c1 11156 < clt 11295 2c2 12321 3c3 12322 |
| 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 df-3 12330 |
| This theorem is referenced by: 1le3 12478 fztpval 13626 fvf1tp 13829 expnass 14247 tpf1ofv1 14536 tpfo 14539 s4fv1 14935 f1oun2prg 14956 sin01gt0 16226 rpnnen2lem3 16252 rpnnen2lem9 16258 3prm 16731 6nprm 17147 7prm 17148 9nprm 17150 13prm 17153 19prm 17155 prmlem2 17157 37prm 17158 43prm 17159 139prm 17161 163prm 17162 631prm 17164 basendxnmulrndx 17339 basendxnmulrndxOLD 17340 opprbasOLD 20342 log2cnv 26987 cxploglim2 27022 2lgslem3 27448 dchrvmasumlem2 27542 pntibndlem1 27633 tgcgr4 28539 axlowdimlem16 28972 usgrexmpldifpr 29275 upgr3v3e3cycl 30199 upgr4cycl4dv4e 30204 konigsberglem2 30272 konigsberglem3 30273 konigsberglem5 30275 frgrogt3nreg 30416 ex-dif 30442 ex-pss 30447 ex-res 30460 evl1deg3 33603 2sqr3minply 33791 aks4d1p1p3 42070 aks4d1p1p2 42071 aks4d1p1p4 42072 aks4d1p3 42079 aks5lem8 42202 acos1half 42388 rabren3dioph 42826 jm2.23 43008 mnringbasedOLD 44231 stoweidlem34 46049 stoweidlem42 46057 smfmullem4 46809 fmtno4prmfac193 47560 3ndvds4 47582 127prm 47586 nnsum4primesodd 47783 nnsum4primesoddALTV 47784 usgrexmpl1lem 47980 usgrexmpl2lem 47985 usgrexmpl2nb1 47991 usgrexmpl2nb3 47993 usgrexmpl2trifr 47996 gpg5grlic 48047 sepfsepc 48825 |
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