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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 10721 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
2 | eqcom 2830 | . . . . 5 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
3 | 2 | orbi1i 910 | . . . 4 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 = 𝐵 ∨ 𝐴 < 𝐵)) |
4 | orcom 866 | . . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
5 | 3, 4 | bitri 277 | . . 3 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) |
6 | axlttri 10714 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) | |
7 | 6 | ancoms 461 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) |
8 | 7 | con2bid 357 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ ¬ 𝐵 < 𝐴)) |
9 | 5, 8 | syl5rbbr 288 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < 𝐴 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
10 | 1, 9 | bitrd 281 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ℝcr 10538 < clt 10677 ≤ cle 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-pre-lttri 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 |
This theorem is referenced by: ltle 10731 leltne 10732 lelttr 10733 ltletr 10734 letr 10736 leid 10738 ltlen 10743 leloei 10759 leloed 10785 lemul1 11494 lemul1a 11496 squeeze0 11545 fimaxreOLD 11587 sup3 11600 nn0ge0 11925 nn0sub 11950 elnn0z 11997 xlemul1a 12684 modfzo0difsn 13314 om2uzlti 13321 om2uzlt2i 13322 sqlecan 13574 discr 13604 facdiv 13650 facwordi 13652 resqrex 14612 sqrt2irr 15604 lcmf 15979 ge2nprmge4 16047 efgsfo 18867 efgred 18876 itg2mulc 24350 itgabs 24437 dgrlt 24858 sinq12ge0 25096 sineq0 25111 cxpge0 25268 cxplea 25281 cxple2 25282 cxple2a 25284 cxpcn3lem 25330 cxpcn3 25331 cxpaddlelem 25334 cxpaddle 25335 ang180lem3 25391 atanlogaddlem 25493 rlimcnp2 25546 jensen 25568 amgm 25570 htthlem 28696 hiidge0 28877 staddi 30025 stadd3i 30027 poimirlem28 34922 itgaddnclem2 34953 itgabsnc 34963 pellfund14gap 39491 sineq0ALT 41278 icccncfext 42177 ltnltne 43506 iccpartnel 43605 odz2prm2pw 43732 evenltle 43889 gbowge7 43935 bgoldbtbndlem1 43977 elfzolborelfzop1 44581 |
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