Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11699 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1 | ltp1i 11532 | . 2 ⊢ 2 < (2 + 1) |
3 | df-3 11689 | . 2 ⊢ 3 = (2 + 1) | |
4 | 2, 3 | breqtrri 5084 | 1 ⊢ 2 < 3 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5057 (class class class)co 7145 1c1 10526 + caddc 10528 < clt 10663 2c2 11680 3c3 11681 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-2 11688 df-3 11689 |
This theorem is referenced by: 1lt3 11798 2lt4 11800 2lt6 11809 2lt7 11815 2lt8 11822 2lt9 11830 3halfnz 12049 2lt10 12224 uzuzle23 12277 uz3m2nn 12279 fztpval 12957 expnass 13558 s4fv2 14247 f1oun2prg 14267 caucvgrlem 15017 cos01gt0 15532 3lcm2e6 16060 5prm 16430 11prm 16436 17prm 16438 23prm 16440 83prm 16444 317prm 16447 4001lem4 16465 plusgndxnmulrndx 16605 rngstr 16607 oppradd 19309 cnfldstr 20475 cnfldfun 20485 2logb9irr 25300 2logb3irr 25302 log2le1 25455 chtub 25715 bpos1 25786 bposlem6 25792 chto1ub 25979 dchrvmasumiflem1 26004 istrkg3ld 26174 tgcgr4 26244 axlowdimlem2 26656 axlowdimlem16 26670 axlowdimlem17 26671 axlowdim 26674 usgrexmpldifpr 26967 upgr3v3e3cycl 27886 konigsbergiedgw 27954 konigsberglem1 27958 konigsberglem2 27959 konigsberglem3 27960 ex-pss 28134 ex-res 28147 ex-fv 28149 ex-fl 28153 ex-mod 28155 prodfzo03 31773 cnndvlem1 33773 poimirlem9 34782 rabren3dioph 39290 jm2.20nn 39472 wallispilem4 42230 fourierdlem87 42355 smfmullem4 42946 257prm 43600 31prm 43637 9fppr8 43779 fpprel2 43783 nnsum3primes4 43830 nnsum3primesgbe 43834 nnsum3primesle9 43836 nnsum4primesodd 43838 nnsum4primesoddALTV 43839 tgoldbach 43859 zlmodzxznm 44480 zlmodzxzldeplem 44481 |
Copyright terms: Public domain | W3C validator |