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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 11339 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
| 2 | axlttri 11332 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) | |
| 3 | 2 | ancoms 458 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) |
| 4 | 3 | con2bid 354 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ ¬ 𝐵 < 𝐴)) |
| 5 | eqcom 2744 | . . . . 5 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
| 6 | 5 | orbi1i 914 | . . . 4 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 = 𝐵 ∨ 𝐴 < 𝐵)) |
| 7 | orcom 871 | . . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
| 8 | 6, 7 | bitri 275 | . . 3 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) |
| 9 | 4, 8 | bitr3di 286 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < 𝐴 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
| 10 | 1, 9 | bitrd 279 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 < clt 11295 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-pre-lttri 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: ltle 11349 leltne 11350 lelttr 11351 ltletr 11353 letr 11355 leid 11357 ltlen 11362 leloei 11378 leloed 11404 lemul1 12119 lemul1a 12121 squeeze0 12171 sup3 12225 nn0ge0 12551 nn0sub 12576 elnn0z 12626 xlemul1a 13330 modfzo0difsn 13984 om2uzlti 13991 om2uzlt2i 13992 sqlecan 14248 discr 14279 facdiv 14326 facwordi 14328 resqrex 15289 sqrt2irr 16285 lcmf 16670 ge2nprmge4 16738 efgsfo 19757 efgred 19766 itg2mulc 25782 itgabs 25870 dgrlt 26306 sinq12ge0 26550 sineq0 26566 cxpge0 26725 cxplea 26738 cxple2 26739 cxple2a 26741 cxpcn3lem 26790 cxpcn3 26791 cxpaddlelem 26794 cxpaddle 26795 ang180lem3 26854 atanlogaddlem 26956 rlimcnp2 27009 jensen 27032 amgm 27034 htthlem 30936 hiidge0 31117 staddi 32265 stadd3i 32267 2exple2exp 32834 poimirlem28 37655 itgaddnclem2 37686 itgabsnc 37696 sn-sup3d 42502 pellfund14gap 42898 sineq0ALT 44957 icccncfext 45902 ltnltne 47311 iccpartnel 47425 odz2prm2pw 47550 evenltle 47704 gbowge7 47750 bgoldbtbndlem1 47792 elfzolborelfzop1 48436 |
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