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Mirrors > Home > MPE Home > Th. List > lttr | Structured version Visualization version GIF version |
Description: Alias for axlttrn 10148, for naming consistency with lttri 10201. New proofs should generally use this instead of ax-pre-lttrn 10049. (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 10148 | 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 1054 ∈ wcel 2030 class class class wbr 4685 ℝcr 9973 < clt 10112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-pre-lttrn 10049 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 |
This theorem is referenced by: ltso 10156 lelttr 10166 ltletr 10167 lttri 10201 lttrd 10236 lt2sub 10564 mulgt1 10920 recgt1i 10958 recreclt 10960 sup2 11017 nnge1 11084 recnz 11490 gtndiv 11492 xrlttr 12011 fzo1fzo0n0 12558 flflp1 12648 1mod 12742 seqf1olem1 12880 expnbnd 13033 expnlbnd 13034 swrd2lsw 13741 2swrd2eqwrdeq 13742 sin01gt0 14964 cos01gt0 14965 p1modz1 15034 ltoddhalfle 15132 nno 15145 dvdsnprmd 15450 chfacfscmul0 20711 chfacfpmmul0 20715 iscmet3lem1 23135 bcthlem4 23170 bcthlem5 23171 ivthlem2 23267 ovolicc2lem3 23333 mbfaddlem 23472 reeff1olem 24245 logdivlti 24411 logblog 24575 ftalem2 24845 chtub 24982 bclbnd 25050 efexple 25051 bposlem1 25054 lgsquadlem2 25151 pntlem3 25343 axlowdimlem16 25882 pthdlem1 26718 wwlksnredwwlkn 26858 clwwlkel 27009 clwwlknonex2lem2 27083 frgrogt3nreg 27384 poimirlem2 33541 stoweidlem34 40569 m1mod0mod1 41664 smonoord 41666 sbgoldbalt 41994 bgoldbtbndlem3 42020 bgoldbtbndlem4 42021 tgoldbach 42030 tgoldbachOLD 42037 difmodm1lt 42642 regt1loggt0 42655 rege1logbrege0 42677 dignn0flhalflem1 42734 |
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