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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 8117 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
| 2 | rexr 8117 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
| 3 | xrlenlt 8136 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
| 4 | 1, 2, 3 | syl2an 289 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
| Colors of variables: wff set class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2175 class class class wbr 4043 ℝcr 7923 ℝ*cxr 8105 < clt 8106 ≤ cle 8107 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-br 4044 df-opab 4105 df-xp 4680 df-cnv 4682 df-xr 8110 df-le 8112 |
| This theorem is referenced by: letri3 8152 ltleletr 8153 letr 8154 leid 8155 eqlelt 8158 ltle 8159 lelttr 8160 ltletr 8161 lenlti 8172 lenltd 8189 lemul1 8665 msqge0 8688 mulge0 8691 ltleap 8704 recgt0 8922 lediv1 8941 dfinfre 9028 nnge1 9058 nnnlt1 9061 avgle1 9277 avgle2 9278 nn0nlt0 9320 zltnle 9417 zleloe 9418 zdcle 9448 recnz 9465 btwnnz 9466 prime 9471 fznlem 10162 nelfzo 10273 fzonlt0 10289 qltnle 10384 bcval4 10895 ccatsymb 11056 resqrexlemgt0 11302 climge0 11607 infpnlem1 12653 efle 15219 logleb 15318 cxple 15360 cxple3 15364 lgsval2lem 15458 lgsneg 15472 lgsdilem 15475 gausslemma2dlem1a 15506 gausslemma2dlem3 15511 supfz 15972 inffz 15973 |
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