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Mirrors > Home > ILE Home > Th. List > lttr | GIF version |
Description: Alias for axlttrn 7181, for naming consistency with lttri 7215. New proofs should generally use this instead of ax-pre-lttrn 7090. (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 7181 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 ∈ wcel 1433 class class class wbr 3785 ℝcr 6980 < clt 7153 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-lttrn 7090 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-ltxr 7158 |
This theorem is referenced by: ltso 7189 ltleletr 7193 ltnsym 7197 lttri 7215 lttrd 7235 lt2add 7549 lt2sub 7564 mulgt1 7941 recgt1i 7976 recreclt 7978 nnge1 8062 recnz 8440 gtndiv 8442 xrlttr 8870 fzo1fzo0n0 9192 expnbnd 9596 expnlbnd 9597 ltoddhalfle 10293 nno 10306 dvdsnprmd 10507 |
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