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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 12383 | . 2 ⊢ 1 < 2 | |
2 | 2lt10 12815 | . 2 ⊢ 2 < ;10 | |
3 | 1re 11214 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 12286 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 10re 12696 | . . 3 ⊢ ;10 ∈ ℝ | |
6 | 3, 4, 5 | lttri 11340 | . 2 ⊢ ((1 < 2 ∧ 2 < ;10) → 1 < ;10) |
7 | 1, 2, 6 | mp2an 691 | 1 ⊢ 1 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5149 0cc0 11110 1c1 11111 < clt 11248 2c2 12267 ;cdc 12677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-dec 12678 |
This theorem is referenced by: 0.999... 15827 3dvds 16274 11prm 17048 13prm 17049 17prm 17050 19prm 17051 23prm 17052 37prm 17054 43prm 17055 83prm 17056 139prm 17057 163prm 17058 317prm 17059 631prm 17060 2503prm 17073 basendxltplendx 17314 basendxnocndx 17328 basendxltdsndx 17333 basendxltunifndx 17343 slotsbhcdif 17360 slotsbhcdifOLD 17361 oppcbasOLD 17664 rescbasOLD 17777 rescabsOLD 17783 catstr 17909 odubasOLD 18245 isposixOLD 18279 cnfldfunALTOLD 20958 znbas2OLD 21093 thlbasOLD 21250 opsrbasOLD 21607 tuslemOLD 23772 tmslemOLD 23991 log2ub 26454 slotsinbpsd 27692 slotslnbpsd 27693 trkgstr 27695 ttgbasOLD 28131 eengstr 28238 basendxltedgfndx 28253 baseltedgfOLD 28254 hgt750lemd 33660 hgt750lem 33663 hgt750lem2 33664 hgt750leme 33670 tgoldbachgnn 33671 3lexlogpow5ineq1 40919 257prm 46229 fmtno4prmfac193 46241 fmtno5nprm 46251 139prmALT 46264 127prm 46267 tgblthelfgott 46483 tgoldbach 46485 |
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