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Mirrors > Home > MPE Home > Th. List > leloe | Structured version Visualization version GIF version |
Description: 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by NM, 13-May-1999.) |
Ref | Expression |
---|---|
leloe | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lenlt 10707 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
2 | eqcom 2825 | . . . . 5 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
3 | 2 | orbi1i 907 | . . . 4 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 = 𝐵 ∨ 𝐴 < 𝐵)) |
4 | orcom 864 | . . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
5 | 3, 4 | bitri 276 | . . 3 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) |
6 | axlttri 10700 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) | |
7 | 6 | ancoms 459 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) |
8 | 7 | con2bid 356 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ ¬ 𝐵 < 𝐴)) |
9 | 5, 8 | syl5rbbr 287 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < 𝐴 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
10 | 1, 9 | bitrd 280 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ℝcr 10524 < clt 10663 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-pre-lttri 10599 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 |
This theorem is referenced by: ltle 10717 leltne 10718 lelttr 10719 ltletr 10720 letr 10722 leid 10724 ltlen 10729 leloei 10745 leloed 10771 lemul1 11480 lemul1a 11482 squeeze0 11531 fimaxreOLD 11573 sup3 11586 nn0ge0 11910 nn0sub 11935 elnn0z 11982 xlemul1a 12669 modfzo0difsn 13299 om2uzlti 13306 om2uzlt2i 13307 sqlecan 13559 discr 13589 facdiv 13635 facwordi 13637 resqrex 14598 sqrt2irr 15590 lcmf 15965 ge2nprmge4 16033 efgsfo 18794 efgred 18803 itg2mulc 24275 itgabs 24362 dgrlt 24783 sinq12ge0 25021 sineq0 25036 cxpge0 25193 cxplea 25206 cxple2 25207 cxple2a 25209 cxpcn3lem 25255 cxpcn3 25256 cxpaddlelem 25259 cxpaddle 25260 ang180lem3 25316 atanlogaddlem 25418 rlimcnp2 25471 jensen 25493 amgm 25495 htthlem 28621 hiidge0 28802 staddi 29950 stadd3i 29952 poimirlem28 34801 itgaddnclem2 34832 itgabsnc 34842 pellfund14gap 39362 sineq0ALT 41148 icccncfext 42046 ltnltne 43376 iccpartnel 43475 odz2prm2pw 43602 evenltle 43759 gbowge7 43805 bgoldbtbndlem1 43847 elfzolborelfzop1 44502 |
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