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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 11802 | . 2 ⊢ 1 < 2 | |
2 | 2lt3 11803 | . 2 ⊢ 2 < 3 | |
3 | 1re 10635 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 11705 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 3re 11711 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10760 | . 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 5058 1c1 10532 < clt 10669 2c2 11686 3c3 11687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-2 11694 df-3 11695 |
This theorem is referenced by: 1le3 11843 fztpval 12963 expnass 13564 s4fv1 14252 f1oun2prg 14273 sin01gt0 15537 rpnnen2lem3 15563 rpnnen2lem9 15569 3prm 16032 6nprm 16437 7prm 16438 9nprm 16440 13prm 16443 19prm 16445 prmlem2 16447 37prm 16448 43prm 16449 139prm 16451 163prm 16452 631prm 16454 basendxnmulrndx 16612 ressmulr 16619 opprbas 19373 log2cnv 25516 cxploglim2 25550 2lgslem3 25974 dchrvmasumlem2 26068 pntibndlem1 26159 tgcgr4 26311 axlowdimlem16 26737 usgrexmpldifpr 27034 upgr3v3e3cycl 27953 upgr4cycl4dv4e 27958 konigsberglem2 28026 konigsberglem3 28027 konigsberglem5 28029 frgrogt3nreg 28170 ex-dif 28196 ex-pss 28201 ex-res 28214 rabren3dioph 39405 jm2.23 39586 stoweidlem34 42313 stoweidlem42 42321 smfmullem4 43063 fmtno4prmfac193 43729 3ndvds4 43752 127prm 43757 nnsum4primesodd 43955 nnsum4primesoddALTV 43956 |
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