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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 11966 | . 2 ⊢ 1 < 2 | |
2 | 2lt10 12396 | . 2 ⊢ 2 < ;10 | |
3 | 1re 10798 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 11869 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 10re 12277 | . . 3 ⊢ ;10 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10923 | . 2 ⊢ ((1 < 2 ∧ 2 < ;10) → 1 < ;10) |
7 | 1, 2, 6 | mp2an 692 | 1 ⊢ 1 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5039 0cc0 10694 1c1 10695 < clt 10832 2c2 11850 ;cdc 12258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-dec 12259 |
This theorem is referenced by: 0.999... 15408 3dvds 15855 11prm 16631 13prm 16632 17prm 16633 19prm 16634 23prm 16635 37prm 16637 43prm 16638 83prm 16639 139prm 16640 163prm 16641 317prm 16642 631prm 16643 2503prm 16656 ressle 16856 ressunif 16863 ressds 16871 resshom 16876 ressco 16877 slotsbhcdif 16878 oppcbas 17176 rescbas 17288 rescabs 17293 catstr 17419 odubas 17753 isposix 17786 cnfldfun 20329 znbas2 20458 thlbas 20612 opsrbas 20961 tuslem 23118 tmslem 23334 log2ub 25786 trkgstr 26489 ttgbas 26922 eengstr 27025 baseltedgf 27039 hgt750lemd 32294 hgt750lem 32297 hgt750lem2 32298 hgt750leme 32304 tgoldbachgnn 32305 3lexlogpow5ineq1 39745 257prm 44629 fmtno4prmfac193 44641 fmtno5nprm 44651 139prmALT 44664 127prm 44667 tgblthelfgott 44883 tgoldbach 44885 |
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