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Mirrors > Home > MPE Home > Th. List > lttr | Structured version Visualization version GIF version |
Description: Alias for axlttrn 10702, for naming consistency with lttri 10755. New proofs should generally use this instead of ax-pre-lttrn 10601. (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 10702 | 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 1079 ∈ wcel 2105 class class class wbr 5058 ℝcr 10525 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-ltxr 10669 |
This theorem is referenced by: ltso 10710 lelttr 10720 ltletr 10721 lttri 10755 lttrd 10790 lt2sub 11127 mulgt1 11488 recgt1i 11526 recreclt 11528 sup2 11586 nnge1 11654 recnz 12046 gtndiv 12048 xrlttr 12523 fzo1fzo0n0 13078 flflp1 13167 1mod 13261 seqf1olem1 13399 expnbnd 13583 expnlbnd 13584 swrd2lsw 14304 2swrd2eqwrdeq 14305 sin01gt0 15533 cos01gt0 15534 p1modz1 15604 ltoddhalfle 15700 nno 15723 dvdsnprmd 16024 chfacfscmul0 21396 chfacfpmmul0 21400 iscmet3lem1 23823 bcthlem4 23859 bcthlem5 23860 ivthlem2 23982 ovolicc2lem3 24049 mbfaddlem 24190 reeff1olem 24963 logdivlti 25130 ftalem2 25579 chtub 25716 bclbnd 25784 efexple 25785 bposlem1 25788 lgsquadlem2 25885 pntlem3 26113 axlowdimlem16 26671 pthdlem1 27475 wwlksnredwwlkn 27601 clwwlkel 27753 clwwlknonex2lem2 27815 frgrogt3nreg 28104 poimirlem2 34776 stoweidlem34 42200 m1mod0mod1 43410 smonoord 43412 sbgoldbalt 43793 bgoldbtbndlem3 43819 bgoldbtbndlem4 43820 tgoldbach 43829 difmodm1lt 44480 regt1loggt0 44494 rege1logbrege0 44516 dignn0flhalflem1 44573 |
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