![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > lttr | GIF version |
Description: Alias for axlttrn 8029, for naming consistency with lttri 8065. New proofs should generally use this instead of ax-pre-lttrn 7928. (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 8029 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 ∈ wcel 2148 class class class wbr 4005 ℝcr 7813 < clt 7995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-pre-lttrn 7928 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-pnf 7997 df-mnf 7998 df-ltxr 8000 |
This theorem is referenced by: ltso 8038 ltleletr 8042 ltnsym 8046 lttri 8065 lttrd 8086 lt2add 8405 lt2sub 8420 mulgt1 8823 recgt1i 8858 recreclt 8860 nnge1 8945 recnz 9349 gtndiv 9351 xrlttr 9798 fzo1fzo0n0 10186 expnbnd 10647 expnlbnd 10648 sin01gt0 11772 cos01gt0 11773 p1modz1 11804 ltoddhalfle 11901 nno 11914 dvdsnprmd 12128 reeff1olem 14353 logdivlti 14463 |
Copyright terms: Public domain | W3C validator |