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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 12340 | . . 3 ⊢ 2 ∈ ℝ | |
| 2 | 1 | ltp1i 12172 | . 2 ⊢ 2 < (2 + 1) |
| 3 | df-3 12330 | . 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 7431 1c1 11156 + caddc 11158 < 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: 1lt3 12439 2lt4 12441 2lt6 12450 2lt7 12456 2lt8 12463 2lt9 12471 3halfnz 12697 2lt10 12871 uzuzle23 12931 uz3m2nn 12933 fztpval 13626 fvf1tp 13829 expnass 14247 hash3tpde 14532 tpf1ofv2 14537 tpfo 14539 s4fv2 14936 f1oun2prg 14956 caucvgrlem 15709 cos01gt0 16227 3lcm2e6 16769 5prm 17146 11prm 17152 17prm 17154 23prm 17156 83prm 17160 317prm 17163 4001lem4 17181 plusgndxnmulrndx 17341 rngstr 17342 slotsdifunifndx 17445 oppraddOLD 20344 cnfldstr 21366 cnfldstrOLD 21381 cnfldfunALTOLDOLD 21393 2logb9irr 26838 2logb3irr 26840 log2le1 26993 chtub 27256 bpos1 27327 bposlem6 27333 chto1ub 27520 dchrvmasumiflem1 27545 istrkg3ld 28469 tgcgr4 28539 axlowdimlem2 28958 axlowdimlem16 28972 axlowdimlem17 28973 axlowdim 28976 usgrexmpldifpr 29275 upgr3v3e3cycl 30199 konigsbergiedgw 30267 konigsberglem1 30271 konigsberglem2 30272 konigsberglem3 30273 ex-pss 30447 ex-res 30460 ex-fv 30462 ex-fl 30466 ex-mod 30468 evl1deg3 33603 2sqr3minply 33791 prodfzo03 34618 cnndvlem1 36538 poimirlem9 37636 3lexlogpow2ineq1 42059 aks4d1p1p6 42074 aks4d1p1p5 42076 2ap1caineq 42146 rabren3dioph 42826 jm2.20nn 43009 mnringaddgdOLD 44237 wallispilem4 46083 fourierdlem87 46208 smfmullem4 46809 257prm 47548 31prm 47584 9fppr8 47724 fpprel2 47728 nnsum3primes4 47775 nnsum3primesgbe 47779 nnsum3primesle9 47781 nnsum4primesodd 47783 nnsum4primesoddALTV 47784 tgoldbach 47804 cycl3grtri 47914 usgrexmpl1lem 47980 usgrexmpl2lem 47985 usgrexmpl2nb2 47992 usgrexmpl2nb3 47993 usgrexmpl2trifr 47996 gpg3nbgrvtx0 48032 gpg3kgrtriexlem1 48039 gpg5grlic 48047 zlmodzxznm 48414 zlmodzxzldeplem 48415 sepfsepc 48825 |
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