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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 7874 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | |
2 | lenlt 7864 | . 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 1481 class class class wbr 3937 ℝcr 7643 < clt 7824 ≤ cle 7825 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 ax-pre-lttrn 7758 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 |
This theorem is referenced by: ltlei 7889 ltled 7905 ltleap 8418 lep1 8627 lem1 8629 letrp1 8630 ltmul12a 8642 bndndx 9000 nn0ge0 9026 zletric 9122 zlelttric 9123 zltnle 9124 zleloe 9125 zdcle 9151 uzind 9186 fnn0ind 9191 eluz2b2 9424 rpge0 9483 zltaddlt1le 9820 difelfznle 9943 elfzouz2 9969 elfzo0le 9993 fzosplitprm1 10042 fzostep1 10045 qletric 10052 qlelttric 10053 qltnle 10054 expgt1 10362 expnlbnd2 10448 faclbnd 10519 caucvgrelemcau 10784 resqrexlemdecn 10816 mulcn2 11113 efcllemp 11401 sin01bnd 11500 cos01bnd 11501 sin01gt0 11504 cos01gt0 11505 absef 11512 efieq1re 11514 nn0o 11640 sincosq1lem 12954 tangtx 12967 |
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