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Mirrors > Home > ILE Home > Th. List > ltle | GIF version |
Description: 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
ltle | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnsym 7818 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | |
2 | lenlt 7808 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
3 | 1, 2 | sylibrd 168 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 1465 class class class wbr 3899 ℝcr 7587 < clt 7768 ≤ cle 7769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 ax-pre-lttrn 7702 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: ltlei 7833 ltled 7849 ltleap 8362 lep1 8571 lem1 8573 letrp1 8574 ltmul12a 8586 bndndx 8944 nn0ge0 8970 zletric 9066 zlelttric 9067 zltnle 9068 zleloe 9069 zdcle 9095 uzind 9130 fnn0ind 9135 eluz2b2 9365 rpge0 9422 zltaddlt1le 9757 difelfznle 9880 elfzouz2 9906 elfzo0le 9930 fzosplitprm1 9979 fzostep1 9982 qletric 9989 qlelttric 9990 qltnle 9991 expgt1 10299 expnlbnd2 10385 faclbnd 10455 caucvgrelemcau 10720 resqrexlemdecn 10752 mulcn2 11049 efcllemp 11291 sin01bnd 11391 cos01bnd 11392 sin01gt0 11395 cos01gt0 11396 absef 11403 efieq1re 11405 nn0o 11531 sincosq1lem 12833 tangtx 12846 |
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