![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10455 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
2 | eqcom 2785 | . . . . 5 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
3 | 2 | orbi1i 900 | . . . 4 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 = 𝐵 ∨ 𝐴 < 𝐵)) |
4 | orcom 859 | . . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
5 | 3, 4 | bitri 267 | . . 3 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) |
6 | axlttri 10448 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) | |
7 | 6 | ancoms 452 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵))) |
8 | 7 | con2bid 346 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ ¬ 𝐵 < 𝐴)) |
9 | 5, 8 | syl5rbbr 278 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐵 < 𝐴 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
10 | 1, 9 | bitrd 271 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 class class class wbr 4886 ℝcr 10271 < clt 10411 ≤ cle 10412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-pre-lttri 10346 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 |
This theorem is referenced by: ltle 10465 leltne 10466 lelttr 10467 ltletr 10468 letr 10470 leid 10472 ltlen 10477 leloei 10493 leloed 10519 lemul1 11229 lemul1a 11231 squeeze0 11280 fimaxre 11322 sup3 11334 nn0ge0 11669 nn0sub 11694 elnn0z 11741 xlemul1a 12430 modfzo0difsn 13061 om2uzlti 13068 om2uzlt2i 13069 sqlecan 13290 discr 13320 facdiv 13392 facwordi 13394 resqrex 14398 sqrt2irr 15382 lcmf 15752 efgsfo 18537 efgred 18547 itg2mulc 23951 itgabs 24038 dgrlt 24459 sinq12ge0 24698 sineq0 24711 cxpge0 24866 cxplea 24879 cxple2 24880 cxple2a 24882 cxpcn3lem 24928 cxpcn3 24929 cxpaddlelem 24932 cxpaddle 24933 ang180lem3 24989 atanlogaddlem 25091 rlimcnp2 25145 jensen 25167 amgm 25169 htthlem 28346 hiidge0 28527 staddi 29677 stadd3i 29679 poimirlem28 34063 itgaddnclem2 34094 itgabsnc 34104 pellfund14gap 38411 sineq0ALT 40106 icccncfext 41028 ltnltne 42341 iccpartnel 42406 odz2prm2pw 42496 evenltle 42651 gbowge7 42676 bgoldbtbndlem1 42718 elfzolborelfzop1 43324 |
Copyright terms: Public domain | W3C validator |