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Mirrors > Home > ILE Home > Th. List > le2tri3i | 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 7125 | . . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 ≤ 𝐴) |
5 | 3, 1 | letri3i 7116 | . . . . . 6 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)) |
6 | 5 | biimpri 124 | . . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐴 = 𝐵) |
7 | 4, 6 | sylan2 270 | . . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) → 𝐴 = 𝐵) |
8 | 7 | 3impb 1100 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐵) |
9 | 2, 3, 1 | letri 7125 | . . . . . 6 ⊢ ((𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐶 ≤ 𝐵) |
10 | 1, 2 | letri3i 7116 | . . . . . . 7 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
11 | 10 | biimpri 124 | . . . . . 6 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐵 = 𝐶) |
12 | 9, 11 | sylan2 270 | . . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ (𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐶) |
13 | 12 | 3impb 1100 | . . . 4 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐶) |
14 | 13 | 3comr 1112 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 = 𝐶) |
15 | 3, 1, 2 | letri 7125 | . . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶) |
16 | 3, 2 | letri3i 7116 | . . . . . . 7 ⊢ (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) |
17 | 16 | biimpri 124 | . . . . . 6 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐶) |
18 | 17 | eqcomd 2045 | . . . . 5 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴) |
19 | 15, 18 | sylan 267 | . . . 4 ⊢ (((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴) |
20 | 19 | 3impa 1099 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴) |
21 | 8, 14, 20 | 3jca 1084 | . 2 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴)) |
22 | 3 | eqlei 7111 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵) |
23 | 1 | eqlei 7111 | . . 3 ⊢ (𝐵 = 𝐶 → 𝐵 ≤ 𝐶) |
24 | 2 | eqlei 7111 | . . 3 ⊢ (𝐶 = 𝐴 → 𝐶 ≤ 𝐴) |
25 | 22, 23, 24 | 3anim123i 1089 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴) → (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) |
26 | 21, 25 | impbii 117 | 1 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 ℝcr 6888 ≤ cle 7061 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-apti 6999 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 |
This theorem is referenced by: (None) |
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