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Mirrors > Home > ILE Home > Th. List > lenlt | GIF version |
Description: 'Less than or equal to' expressed in terms of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-May-1999.) |
Ref | Expression |
---|---|
lenlt | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 7596 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 7596 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrlenlt 7614 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | syl2an 284 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1439 class class class wbr 3853 ℝcr 7412 ℝ*cxr 7584 < clt 7585 ≤ cle 7586 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2624 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-br 3854 df-opab 3908 df-xp 4460 df-cnv 4462 df-xr 7589 df-le 7591 |
This theorem is referenced by: letri3 7629 ltleletr 7630 letr 7631 leid 7632 ltle 7635 lelttr 7636 ltletr 7637 lenlti 7648 lenltd 7664 lemul1 8133 msqge0 8156 mulge0 8159 ltleap 8170 recgt0 8374 lediv1 8393 dfinfre 8480 nnge1 8508 nnnlt1 8511 avgle1 8719 avgle2 8720 nn0nlt0 8762 zltnle 8859 zleloe 8860 zdcle 8886 recnz 8902 btwnnz 8903 prime 8908 fznlem 9518 fzonlt0 9641 qltnle 9720 bcval4 10223 resqrexlemgt0 10516 climge0 10776 efler 11052 supfz 12219 inffz 12220 |
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