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| Description: Trichotomy law. (Contributed by NM, 4-Apr-2005.) | 
| Ref | Expression | 
|---|---|
| lelttric | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pm2.1 896 | . 2 ⊢ (¬ 𝐵 < 𝐴 ∨ 𝐵 < 𝐴) | |
| 2 | lenlt 11340 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
| 3 | 2 | orbi1d 916 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴) ↔ (¬ 𝐵 < 𝐴 ∨ 𝐵 < 𝐴))) | 
| 4 | 1, 3 | mpbiri 258 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∈ wcel 2107 class class class wbr 5142 ℝcr 11155 < clt 11296 ≤ cle 11297 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-xr 11300 df-le 11302 | 
| This theorem is referenced by: ltlecasei 11370 fzsplit2 13590 uzsplit 13637 fzospliti 13732 fzouzsplit 13735 discr1 14279 faclbnd 14330 faclbnd4lem1 14333 faclbnd4lem4 14336 dvdslelem 16347 dvdsprmpweqle 16925 icccmplem2 24846 icccmp 24848 bcmono 27322 bpos1lem 27327 bposlem3 27331 bpos 27338 fzsplit3 32796 submateq 33809 lzunuz 42784 jm2.24 42980 fzuntgd 43476 iccpartnel 47430 bgoldbtbnd 47801 tgoldbach 47809 reorelicc 48636 | 
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