|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > le2tri3i | Structured version Visualization version GIF version | ||
| Description: Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.) | 
| Ref | Expression | 
|---|---|
| lt.1 | ⊢ 𝐴 ∈ ℝ | 
| lt.2 | ⊢ 𝐵 ∈ ℝ | 
| lt.3 | ⊢ 𝐶 ∈ ℝ | 
| Ref | Expression | 
|---|---|
| le2tri3i | ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lt.2 | . . . . . 6 ⊢ 𝐵 ∈ ℝ | |
| 2 | lt.3 | . . . . . 6 ⊢ 𝐶 ∈ ℝ | |
| 3 | lt.1 | . . . . . 6 ⊢ 𝐴 ∈ ℝ | |
| 4 | 1, 2, 3 | letri 11391 | . . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 ≤ 𝐴) | 
| 5 | 3, 1 | letri3i 11378 | . . . . . 6 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)) | 
| 6 | 5 | biimpri 228 | . . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐴 = 𝐵) | 
| 7 | 4, 6 | sylan2 593 | . . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) → 𝐴 = 𝐵) | 
| 8 | 7 | 3impb 1114 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐵) | 
| 9 | 2, 3, 1 | letri 11391 | . . . . . 6 ⊢ ((𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐶 ≤ 𝐵) | 
| 10 | 1, 2 | letri3i 11378 | . . . . . . 7 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) | 
| 11 | 10 | biimpri 228 | . . . . . 6 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐵 = 𝐶) | 
| 12 | 9, 11 | sylan2 593 | . . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ (𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐶) | 
| 13 | 12 | 3impb 1114 | . . . 4 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐶) | 
| 14 | 13 | 3comr 1125 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 = 𝐶) | 
| 15 | 3, 1, 2 | letri 11391 | . . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶) | 
| 16 | 3, 2 | letri3i 11378 | . . . . . 6 ⊢ (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) | 
| 17 | 16 | biimpri 228 | . . . . 5 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐶) | 
| 18 | 17 | eqcomd 2742 | . . . 4 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴) | 
| 19 | 15, 18 | stoic3 1775 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴) | 
| 20 | 8, 14, 19 | 3jca 1128 | . 2 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴)) | 
| 21 | 3 | eqlei 11372 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵) | 
| 22 | 1 | eqlei 11372 | . . 3 ⊢ (𝐵 = 𝐶 → 𝐵 ≤ 𝐶) | 
| 23 | 2 | eqlei 11372 | . . 3 ⊢ (𝐶 = 𝐴 → 𝐶 ≤ 𝐴) | 
| 24 | 21, 22, 23 | 3anim123i 1151 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴) → (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) | 
| 25 | 20, 24 | impbii 209 | 1 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ℝcr 11155 ≤ 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-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |