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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 11393 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1 | ltp1i 11219 | . 2 ⊢ 3 < (3 + 1) |
3 | df-4 11378 | . 2 ⊢ 4 = (3 + 1) | |
4 | 2, 3 | breqtrri 4870 | 1 ⊢ 3 < 4 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4843 (class class class)co 6878 1c1 10225 + caddc 10227 < clt 10363 3c3 11369 4c4 11370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-2 11376 df-3 11377 df-4 11378 |
This theorem is referenced by: 2lt4 11495 3lt5 11498 3lt6 11503 3lt7 11509 3lt8 11516 3lt9 11524 3halfnz 11746 3lt10 11922 fz0to4untppr 12697 fldiv4p1lem1div2 12891 bpoly4 15126 ef01bndlem 15250 sin01bnd 15251 flodddiv4 15472 srngfn 16329 cnfldfun 20080 dveflem 24083 tangtx 24599 ppiublem1 25279 bpos1 25360 bposlem2 25362 gausslemma2dlem4 25446 2lgslem3b 25474 2lgslem3d 25476 chebbnd1lem2 25511 chebbnd1lem3 25512 chebbnd1 25513 pntlemb 25638 usgrexmplef 26493 upgr4cycl4dv4e 27529 ex-fl 27832 hlhilsmul 37962 stoweidlem26 40986 stoweid 41023 mod42tp1mod8 42301 nnsum4primes4 42459 nnsum4primesprm 42461 nnsum4primesgbe 42463 nnsum4primesle9 42465 nnsum4primeseven 42470 nnsum4primesevenALTV 42471 wtgoldbnnsum4prm 42472 |
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