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Mirrors > Home > MPE Home > Th. List > 8lt10 | Structured version Visualization version GIF version |
Description: 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
8lt10 | ⊢ 8 < ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8lt9 11406 | . 2 ⊢ 8 < 9 | |
2 | 9lt10 11857 | . 2 ⊢ 9 < ;10 | |
3 | 8re 11289 | . . 3 ⊢ 8 ∈ ℝ | |
4 | 9re 11291 | . . 3 ⊢ 9 ∈ ℝ | |
5 | 10re 11701 | . . 3 ⊢ ;10 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10347 | . 2 ⊢ ((8 < 9 ∧ 9 < ;10) → 8 < ;10) |
7 | 1, 2, 6 | mp2an 710 | 1 ⊢ 8 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4796 0cc0 10120 1c1 10121 < clt 10258 8c8 11260 9c9 11261 ;cdc 11677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-7 11268 df-8 11269 df-9 11270 df-dec 11678 |
This theorem is referenced by: 7lt10 11859 83prm 16024 317prm 16027 1259lem5 16036 2503prm 16041 srads 19380 hgt750lem 31030 hgt750lem2 31031 127prm 42017 |
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