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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 7843 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | |
2 | lenlt 7833 | . 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 1480 class class class wbr 3924 ℝcr 7612 < clt 7793 ≤ cle 7794 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-lttrn 7727 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 |
This theorem is referenced by: ltlei 7858 ltled 7874 ltleap 8387 lep1 8596 lem1 8598 letrp1 8599 ltmul12a 8611 bndndx 8969 nn0ge0 8995 zletric 9091 zlelttric 9092 zltnle 9093 zleloe 9094 zdcle 9120 uzind 9155 fnn0ind 9160 eluz2b2 9390 rpge0 9447 zltaddlt1le 9782 difelfznle 9905 elfzouz2 9931 elfzo0le 9955 fzosplitprm1 10004 fzostep1 10007 qletric 10014 qlelttric 10015 qltnle 10016 expgt1 10324 expnlbnd2 10410 faclbnd 10480 caucvgrelemcau 10745 resqrexlemdecn 10777 mulcn2 11074 efcllemp 11353 sin01bnd 11453 cos01bnd 11454 sin01gt0 11457 cos01gt0 11458 absef 11465 efieq1re 11467 nn0o 11593 sincosq1lem 12895 tangtx 12908 |
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