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Mirrors > Home > MPE Home > Th. List > 2lt3 | Structured version Visualization version GIF version |
Description: 2 is less than 3. (Contributed by NM, 26-Sep-2010.) |
Ref | Expression |
---|---|
2lt3 | ⊢ 2 < 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 12329 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1 | ltp1i 12161 | . 2 ⊢ 2 < (2 + 1) |
3 | df-3 12319 | . 2 ⊢ 3 = (2 + 1) | |
4 | 2, 3 | breqtrri 5170 | 1 ⊢ 2 < 3 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5143 (class class class)co 7413 1c1 11147 + caddc 11149 < clt 11286 2c2 12310 3c3 12311 |
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 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-2 12318 df-3 12319 |
This theorem is referenced by: 1lt3 12428 2lt4 12430 2lt6 12439 2lt7 12445 2lt8 12452 2lt9 12460 3halfnz 12684 2lt10 12858 uzuzle23 12916 uz3m2nn 12918 fztpval 13608 expnass 14217 s4fv2 14898 f1oun2prg 14918 caucvgrlem 15669 cos01gt0 16185 3lcm2e6 16726 5prm 17103 11prm 17109 17prm 17111 23prm 17113 83prm 17117 317prm 17120 4001lem4 17138 plusgndxnmulrndx 17303 rngstr 17304 slotsdifunifndx 17407 oppraddOLD 20319 cnfldstr 21338 cnfldstrOLD 21353 cnfldfunALTOLDOLD 21365 2logb9irr 26817 2logb3irr 26819 log2le1 26972 chtub 27235 bpos1 27306 bposlem6 27312 chto1ub 27499 dchrvmasumiflem1 27524 istrkg3ld 28382 tgcgr4 28452 axlowdimlem2 28871 axlowdimlem16 28885 axlowdimlem17 28886 axlowdim 28889 usgrexmpldifpr 29188 upgr3v3e3cycl 30107 konigsbergiedgw 30175 konigsberglem1 30179 konigsberglem2 30180 konigsberglem3 30181 ex-pss 30355 ex-res 30368 ex-fv 30370 ex-fl 30374 ex-mod 30376 evl1deg3 33453 2sqr3minply 33617 prodfzo03 34459 cnndvlem1 36250 poimirlem9 37340 3lexlogpow2ineq1 41767 aks4d1p1p6 41782 aks4d1p1p5 41784 2ap1caineq 41854 rabren3dioph 42506 jm2.20nn 42689 mnringaddgdOLD 43926 wallispilem4 45722 fourierdlem87 45847 smfmullem4 46448 257prm 47166 31prm 47202 9fppr8 47342 fpprel2 47346 nnsum3primes4 47393 nnsum3primesgbe 47397 nnsum3primesle9 47399 nnsum4primesodd 47401 nnsum4primesoddALTV 47402 tgoldbach 47422 zlmodzxznm 47913 zlmodzxzldeplem 47914 sepfsepc 48294 |
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