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Mirrors > Home > MPE Home > Th. List > lttr | Structured version Visualization version GIF version |
Description: Alias for axlttrn 10322, for naming consistency with lttri 10375. New proofs should generally use this instead of ax-pre-lttrn 10223. (Contributed by NM, 10-Mar-2008.) |
Ref | Expression |
---|---|
lttr | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axlttrn 10322 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 ∈ wcel 2139 class class class wbr 4804 ℝcr 10147 < clt 10286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-resscn 10205 ax-pre-lttrn 10223 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-ltxr 10291 |
This theorem is referenced by: ltso 10330 lelttr 10340 ltletr 10341 lttri 10375 lttrd 10410 lt2sub 10738 mulgt1 11094 recgt1i 11132 recreclt 11134 sup2 11191 nnge1 11258 recnz 11664 gtndiv 11666 xrlttr 12186 fzo1fzo0n0 12733 flflp1 12822 1mod 12916 seqf1olem1 13054 expnbnd 13207 expnlbnd 13208 swrd2lsw 13916 2swrd2eqwrdeq 13917 sin01gt0 15139 cos01gt0 15140 p1modz1 15209 ltoddhalfle 15307 nno 15320 dvdsnprmd 15625 chfacfscmul0 20885 chfacfpmmul0 20889 iscmet3lem1 23309 bcthlem4 23344 bcthlem5 23345 ivthlem2 23441 ovolicc2lem3 23507 mbfaddlem 23646 reeff1olem 24419 logdivlti 24586 logblog 24750 ftalem2 25020 chtub 25157 bclbnd 25225 efexple 25226 bposlem1 25229 lgsquadlem2 25326 pntlem3 25518 axlowdimlem16 26057 pthdlem1 26893 wwlksnredwwlkn 27034 clwwlkel 27196 clwwlknonex2lem2 27278 frgrogt3nreg 27586 poimirlem2 33742 stoweidlem34 40772 m1mod0mod1 41867 smonoord 41869 sbgoldbalt 42197 bgoldbtbndlem3 42223 bgoldbtbndlem4 42224 tgoldbach 42233 tgoldbachOLD 42240 difmodm1lt 42845 regt1loggt0 42858 rege1logbrege0 42880 dignn0flhalflem1 42937 |
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