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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 11797 | . 2 ⊢ 1 < 2 | |
2 | 2lt3 11798 | . 2 ⊢ 2 < 3 | |
3 | 1re 10630 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 11700 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 3re 11706 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10755 | . 2 ⊢ ((1 < 2 ∧ 2 < 3) → 1 < 3) |
7 | 1, 2, 6 | mp2an 688 | 1 ⊢ 1 < 3 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5058 1c1 10527 < clt 10664 2c2 11681 3c3 11682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-2 11689 df-3 11690 |
This theorem is referenced by: 1le3 11838 fztpval 12959 expnass 13560 s4fv1 14248 f1oun2prg 14269 sin01gt0 15533 rpnnen2lem3 15559 rpnnen2lem9 15565 3prm 16028 6nprm 16433 7prm 16434 9nprm 16436 13prm 16439 19prm 16441 prmlem2 16443 37prm 16444 43prm 16445 139prm 16447 163prm 16448 631prm 16450 basendxnmulrndx 16608 ressmulr 16615 opprbas 19310 log2cnv 25450 cxploglim2 25484 2lgslem3 25908 dchrvmasumlem2 26002 pntibndlem1 26093 tgcgr4 26245 axlowdimlem16 26671 usgrexmpldifpr 26968 upgr3v3e3cycl 27887 upgr4cycl4dv4e 27892 konigsberglem2 27960 konigsberglem3 27961 konigsberglem5 27963 frgrogt3nreg 28104 ex-dif 28130 ex-pss 28135 ex-res 28148 rabren3dioph 39292 jm2.23 39473 stoweidlem34 42200 stoweidlem42 42208 smfmullem4 42950 fmtno4prmfac193 43582 3ndvds4 43605 127prm 43610 nnsum4primesodd 43808 nnsum4primesoddALTV 43809 |
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