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Mirrors > Home > MPE Home > Th. List > 6lt8 | Structured version Visualization version GIF version |
Description: 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
6lt8 | ⊢ 6 < 8 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6lt7 11372 | . 2 ⊢ 6 < 7 | |
2 | 7lt8 11378 | . 2 ⊢ 7 < 8 | |
3 | 6re 11264 | . . 3 ⊢ 6 ∈ ℝ | |
4 | 7re 11266 | . . 3 ⊢ 7 ∈ ℝ | |
5 | 8re 11268 | . . 3 ⊢ 8 ∈ ℝ | |
6 | 3, 4, 5 | lttri 10326 | . 2 ⊢ ((6 < 7 ∧ 7 < 8) → 6 < 8) |
7 | 1, 2, 6 | mp2an 710 | 1 ⊢ 6 < 8 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4792 < clt 10237 6c6 11237 7c7 11238 8c8 11239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-po 5175 df-so 5176 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-er 7899 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-2 11242 df-3 11243 df-4 11244 df-5 11245 df-6 11246 df-7 11247 df-8 11248 |
This theorem is referenced by: 5lt8 11380 631prm 16007 ipsstr 16197 phlstr 16207 sralem 19350 sravsca 19355 chtub 25107 bpos1 25178 |
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