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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 8079 | . . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 ≤ 𝐴) |
5 | 3, 1 | letri3i 8070 | . . . . . 6 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)) |
6 | 5 | biimpri 133 | . . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) → 𝐴 = 𝐵) |
7 | 4, 6 | sylan2 286 | . . . 4 ⊢ ((𝐴 ≤ 𝐵 ∧ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) → 𝐴 = 𝐵) |
8 | 7 | 3impb 1200 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐵) |
9 | 2, 3, 1 | letri 8079 | . . . . . 6 ⊢ ((𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐶 ≤ 𝐵) |
10 | 1, 2 | letri3i 8070 | . . . . . . 7 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
11 | 10 | biimpri 133 | . . . . . 6 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐵 = 𝐶) |
12 | 9, 11 | sylan2 286 | . . . . 5 ⊢ ((𝐵 ≤ 𝐶 ∧ (𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐵 = 𝐶) |
13 | 12 | 3impb 1200 | . . . 4 ⊢ ((𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝐵 = 𝐶) |
14 | 13 | 3comr 1212 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐵 = 𝐶) |
15 | 3, 1, 2 | letri 8079 | . . . . 5 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶) |
16 | 3, 2 | letri3i 8070 | . . . . . . 7 ⊢ (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) |
17 | 16 | biimpri 133 | . . . . . 6 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐴 = 𝐶) |
18 | 17 | eqcomd 2193 | . . . . 5 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴) |
19 | 15, 18 | sylan 283 | . . . 4 ⊢ (((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴) |
20 | 19 | 3impa 1195 | . . 3 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → 𝐶 = 𝐴) |
21 | 8, 14, 20 | 3jca 1178 | . 2 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴)) |
22 | 3 | eqlei 8065 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵) |
23 | 1 | eqlei 8065 | . . 3 ⊢ (𝐵 = 𝐶 → 𝐵 ≤ 𝐶) |
24 | 2 | eqlei 8065 | . . 3 ⊢ (𝐶 = 𝐴 → 𝐶 ≤ 𝐴) |
25 | 22, 23, 24 | 3anim123i 1185 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴) → (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴)) |
26 | 21, 25 | impbii 126 | 1 ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∧ w3a 979 = wceq 1363 ∈ wcel 2158 class class class wbr 4015 ℝcr 7824 ≤ cle 8007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-apti 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-cnv 4646 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 |
This theorem is referenced by: (None) |
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