![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lttr | Structured version Visualization version GIF version |
Description: Alias for axlttrn 11282, for naming consistency with lttri 11336. New proofs should generally use this instead of ax-pre-lttrn 11181. (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 11282 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5147 ℝcr 11105 < clt 11244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 |
This theorem is referenced by: ltso 11290 lelttr 11300 ltletr 11302 lttri 11336 lttrd 11371 lt2sub 11708 mulgt1 12069 recgt1i 12107 recreclt 12109 sup2 12166 nnge1 12236 recnz 12633 gtndiv 12635 xrlttr 13115 fzo1fzo0n0 13679 flflp1 13768 1mod 13864 seqf1olem1 14003 expnbnd 14191 expnlbnd 14192 swrd2lsw 14899 2swrd2eqwrdeq 14900 sin01gt0 16129 cos01gt0 16130 p1modz1 16200 ltoddhalfle 16300 nno 16321 dvdsnprmd 16623 chfacfscmul0 22351 chfacfpmmul0 22355 iscmet3lem1 24799 bcthlem4 24835 bcthlem5 24836 ivthlem2 24960 ovolicc2lem3 25027 mbfaddlem 25168 reeff1olem 25949 logdivlti 26119 ftalem2 26567 chtub 26704 bclbnd 26772 efexple 26773 bposlem1 26776 lgsquadlem2 26873 pntlem3 27101 axlowdimlem16 28204 pthdlem1 29012 wwlksnredwwlkn 29138 clwwlkel 29288 clwwlknonex2lem2 29350 frgrogt3nreg 29639 poimirlem2 36478 sn-sup2 41338 stoweidlem34 44736 m1mod0mod1 46023 smonoord 46025 sbgoldbalt 46435 bgoldbtbndlem3 46461 bgoldbtbndlem4 46462 tgoldbach 46471 difmodm1lt 47161 regt1loggt0 47175 rege1logbrege0 47197 dignn0flhalflem1 47254 |
Copyright terms: Public domain | W3C validator |