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Mirrors > Home > MPE Home > Th. List > 1lt9 | Structured version Visualization version GIF version |
Description: 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
1lt9 | ⊢ 1 < 9 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2 12435 | . 2 ⊢ 1 < 2 | |
2 | 2lt9 12469 | . 2 ⊢ 2 < 9 | |
3 | 1re 11264 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2re 12338 | . . 3 ⊢ 2 ∈ ℝ | |
5 | 9re 12363 | . . 3 ⊢ 9 ∈ ℝ | |
6 | 3, 4, 5 | lttri 11390 | . 2 ⊢ ((1 < 2 ∧ 2 < 9) → 1 < 9) |
7 | 1, 2, 6 | mp2an 690 | 1 ⊢ 1 < 9 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5153 1c1 11159 < clt 11298 2c2 12319 9c9 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 |
This theorem is referenced by: prmlem2 17122 163prm 17127 2503lem3 17141 basendxlttsetndx 17369 otpsstr 17390 ipostr 18554 symgvalstructOLD 19395 cnfldfunALTOLDOLD 21372 eltpsgOLD 22937 indistpsALTOLD 23008 tuslemOLD 24263 setsmsbasOLD 24473 tngbasOLD 24643 log2tlbnd 26973 hgt750lem 34497 hgt750leme 34504 aks4d1p7 41782 aks4d1p8 41786 fmtno4nprmfac193 47146 2exp340mod341 47305 8exp8mod9 47308 nfermltl8rev 47314 |
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