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Mirrors > Home > MPE Home > Th. List > 1lt10 | Structured version Visualization version GIF version |
Description: 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
1lt10 | ⊢ 1 < ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2 11557 | . 2 ⊢ 1 < 2 | |
2 | 2lt10 11989 | . 2 ⊢ 2 < ;10 | |
3 | 1re 10378 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 11453 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 10re 11868 | . . 3 ⊢ ;10 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10504 | . 2 ⊢ ((1 < 2 ∧ 2 < ;10) → 1 < ;10) |
7 | 1, 2, 6 | mp2an 682 | 1 ⊢ 1 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4888 0cc0 10274 1c1 10275 < clt 10413 2c2 11434 ;cdc 11849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-dec 11850 |
This theorem is referenced by: 0.999... 15020 3dvds 15463 11prm 16224 13prm 16225 17prm 16226 19prm 16227 23prm 16228 37prm 16230 43prm 16231 83prm 16232 139prm 16233 163prm 16234 317prm 16235 631prm 16236 2503prm 16249 ressle 16449 ressds 16463 resshom 16468 ressco 16469 slotsbhcdif 16470 oppcbas 16767 rescbas 16878 rescabs 16882 catstr 17006 isposix 17347 odubas 17523 opsrbas 19879 cnfldfun 20158 znbas2 20287 thlbas 20443 ressunif 22478 tuslem 22483 tmslem 22699 log2ub 25132 trkgstr 25799 ttgbas 26230 eengstr 26333 baseltedgf 26346 hgt750lemd 31332 hgt750lem 31335 hgt750lem2 31336 hgt750leme 31342 tgoldbachgnn 31343 257prm 42504 fmtno4prmfac193 42516 fmtno5nprm 42526 139prmALT 42542 127prm 42546 tgblthelfgott 42738 tgoldbach 42740 |
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