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Mirrors > Home > MPE Home > Th. List > 3lt4 | Structured version Visualization version GIF version |
Description: 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
3lt4 | ⊢ 3 < 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3re 11294 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1 | ltp1i 11127 | . 2 ⊢ 3 < (3 + 1) |
3 | df-4 11281 | . 2 ⊢ 4 = (3 + 1) | |
4 | 2, 3 | breqtrri 4813 | 1 ⊢ 3 < 4 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4786 (class class class)co 6791 1c1 10137 + caddc 10139 < clt 10274 3c3 11271 4c4 11272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-2 11279 df-3 11280 df-4 11281 |
This theorem is referenced by: 2lt4 11398 3lt5 11401 3lt6 11406 3lt7 11412 3lt8 11419 3lt9 11427 3lt10OLD 11436 3halfnz 11656 3lt10 11878 fz0to4untppr 12643 fldiv4p1lem1div2 12837 bpoly4 14989 ef01bndlem 15113 sin01bnd 15114 flodddiv4 15338 srngfn 16209 cnfldfun 19966 dveflem 23955 tangtx 24471 ppiublem1 25141 bpos1 25222 bposlem2 25224 gausslemma2dlem4 25308 2lgslem3b 25336 2lgslem3d 25338 chebbnd1lem2 25373 chebbnd1lem3 25374 chebbnd1 25375 pntlemb 25500 usgrexmplef 26367 upgr4cycl4dv4e 27358 ex-fl 27639 hlhilsmul 37744 stoweidlem26 40753 stoweid 40790 mod42tp1mod8 42040 nnsum4primes4 42198 nnsum4primesprm 42200 nnsum4primesgbe 42202 nnsum4primesle9 42204 nnsum4primeseven 42209 nnsum4primesevenALTV 42210 wtgoldbnnsum4prm 42211 |
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