Proof of Theorem hash3tpde
| Step | Hyp | Ref
| Expression |
| 1 | | hash3tr 14530 |
. 2
⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐}) |
| 2 | | ax-1 6 |
. . . . . . 7
⊢ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) → (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 3 | | 3ianor 1107 |
. . . . . . . . 9
⊢ (¬
(𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ (¬ 𝑎 ≠ 𝑏 ∨ ¬ 𝑎 ≠ 𝑐 ∨ ¬ 𝑏 ≠ 𝑐)) |
| 4 | | nne 2944 |
. . . . . . . . . 10
⊢ (¬
𝑎 ≠ 𝑏 ↔ 𝑎 = 𝑏) |
| 5 | | nne 2944 |
. . . . . . . . . 10
⊢ (¬
𝑎 ≠ 𝑐 ↔ 𝑎 = 𝑐) |
| 6 | | nne 2944 |
. . . . . . . . . 10
⊢ (¬
𝑏 ≠ 𝑐 ↔ 𝑏 = 𝑐) |
| 7 | 4, 5, 6 | 3orbi123i 1157 |
. . . . . . . . 9
⊢ ((¬
𝑎 ≠ 𝑏 ∨ ¬ 𝑎 ≠ 𝑐 ∨ ¬ 𝑏 ≠ 𝑐) ↔ (𝑎 = 𝑏 ∨ 𝑎 = 𝑐 ∨ 𝑏 = 𝑐)) |
| 8 | 3, 7 | bitri 275 |
. . . . . . . 8
⊢ (¬
(𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ (𝑎 = 𝑏 ∨ 𝑎 = 𝑐 ∨ 𝑏 = 𝑐)) |
| 9 | | tpeq1 4742 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑏 → {𝑎, 𝑏, 𝑐} = {𝑏, 𝑏, 𝑐}) |
| 10 | | tpidm12 4755 |
. . . . . . . . . . . . 13
⊢ {𝑏, 𝑏, 𝑐} = {𝑏, 𝑐} |
| 11 | 9, 10 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑏 → {𝑎, 𝑏, 𝑐} = {𝑏, 𝑐}) |
| 12 | 11 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑏 → (𝑉 = {𝑎, 𝑏, 𝑐} ↔ 𝑉 = {𝑏, 𝑐})) |
| 13 | | fveqeq2 6915 |
. . . . . . . . . . . . 13
⊢ (𝑉 = {𝑏, 𝑐} → ((♯‘𝑉) = 3 ↔ (♯‘{𝑏, 𝑐}) = 3)) |
| 14 | | hashprlei 14507 |
. . . . . . . . . . . . . 14
⊢ ({𝑏, 𝑐} ∈ Fin ∧ (♯‘{𝑏, 𝑐}) ≤ 2) |
| 15 | | breq1 5146 |
. . . . . . . . . . . . . . . . 17
⊢
((♯‘{𝑏,
𝑐}) = 3 →
((♯‘{𝑏, 𝑐}) ≤ 2 ↔ 3 ≤
2)) |
| 16 | | 2lt3 12438 |
. . . . . . . . . . . . . . . . . . 19
⊢ 2 <
3 |
| 17 | | 2re 12340 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 2 ∈
ℝ |
| 18 | | 3re 12346 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 3 ∈
ℝ |
| 19 | 17, 18 | ltnlei 11382 |
. . . . . . . . . . . . . . . . . . 19
⊢ (2 < 3
↔ ¬ 3 ≤ 2) |
| 20 | 16, 19 | mpbi 230 |
. . . . . . . . . . . . . . . . . 18
⊢ ¬ 3
≤ 2 |
| 21 | 20 | pm2.21i 119 |
. . . . . . . . . . . . . . . . 17
⊢ (3 ≤ 2
→ (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
| 22 | 15, 21 | biimtrdi 253 |
. . . . . . . . . . . . . . . 16
⊢
((♯‘{𝑏,
𝑐}) = 3 →
((♯‘{𝑏, 𝑐}) ≤ 2 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 23 | 22 | com12 32 |
. . . . . . . . . . . . . . 15
⊢
((♯‘{𝑏,
𝑐}) ≤ 2 →
((♯‘{𝑏, 𝑐}) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 24 | 23 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (({𝑏, 𝑐} ∈ Fin ∧ (♯‘{𝑏, 𝑐}) ≤ 2) → ((♯‘{𝑏, 𝑐}) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 25 | 14, 24 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
((♯‘{𝑏,
𝑐}) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
| 26 | 13, 25 | biimtrdi 253 |
. . . . . . . . . . . 12
⊢ (𝑉 = {𝑏, 𝑐} → ((♯‘𝑉) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 27 | 26 | adantld 490 |
. . . . . . . . . . 11
⊢ (𝑉 = {𝑏, 𝑐} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 28 | 12, 27 | biimtrdi 253 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → (𝑉 = {𝑎, 𝑏, 𝑐} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))) |
| 29 | | tpeq1 4742 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑐 → {𝑎, 𝑏, 𝑐} = {𝑐, 𝑏, 𝑐}) |
| 30 | | tpidm13 4756 |
. . . . . . . . . . . . 13
⊢ {𝑐, 𝑏, 𝑐} = {𝑐, 𝑏} |
| 31 | 29, 30 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑐 → {𝑎, 𝑏, 𝑐} = {𝑐, 𝑏}) |
| 32 | 31 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑐 → (𝑉 = {𝑎, 𝑏, 𝑐} ↔ 𝑉 = {𝑐, 𝑏})) |
| 33 | | fveqeq2 6915 |
. . . . . . . . . . . . 13
⊢ (𝑉 = {𝑐, 𝑏} → ((♯‘𝑉) = 3 ↔ (♯‘{𝑐, 𝑏}) = 3)) |
| 34 | | hashprlei 14507 |
. . . . . . . . . . . . . 14
⊢ ({𝑐, 𝑏} ∈ Fin ∧ (♯‘{𝑐, 𝑏}) ≤ 2) |
| 35 | | breq1 5146 |
. . . . . . . . . . . . . . . . 17
⊢
((♯‘{𝑐,
𝑏}) = 3 →
((♯‘{𝑐, 𝑏}) ≤ 2 ↔ 3 ≤
2)) |
| 36 | 35, 21 | biimtrdi 253 |
. . . . . . . . . . . . . . . 16
⊢
((♯‘{𝑐,
𝑏}) = 3 →
((♯‘{𝑐, 𝑏}) ≤ 2 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 37 | 36 | com12 32 |
. . . . . . . . . . . . . . 15
⊢
((♯‘{𝑐,
𝑏}) ≤ 2 →
((♯‘{𝑐, 𝑏}) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 38 | 37 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (({𝑐, 𝑏} ∈ Fin ∧ (♯‘{𝑐, 𝑏}) ≤ 2) → ((♯‘{𝑐, 𝑏}) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 39 | 34, 38 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
((♯‘{𝑐,
𝑏}) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
| 40 | 33, 39 | biimtrdi 253 |
. . . . . . . . . . . 12
⊢ (𝑉 = {𝑐, 𝑏} → ((♯‘𝑉) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 41 | 40 | adantld 490 |
. . . . . . . . . . 11
⊢ (𝑉 = {𝑐, 𝑏} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 42 | 32, 41 | biimtrdi 253 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑐 → (𝑉 = {𝑎, 𝑏, 𝑐} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))) |
| 43 | | tpeq2 4743 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 𝑐 → {𝑎, 𝑏, 𝑐} = {𝑎, 𝑐, 𝑐}) |
| 44 | | tpidm23 4757 |
. . . . . . . . . . . . 13
⊢ {𝑎, 𝑐, 𝑐} = {𝑎, 𝑐} |
| 45 | 43, 44 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝑐 → {𝑎, 𝑏, 𝑐} = {𝑎, 𝑐}) |
| 46 | 45 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑐 → (𝑉 = {𝑎, 𝑏, 𝑐} ↔ 𝑉 = {𝑎, 𝑐})) |
| 47 | | fveqeq2 6915 |
. . . . . . . . . . . . 13
⊢ (𝑉 = {𝑎, 𝑐} → ((♯‘𝑉) = 3 ↔ (♯‘{𝑎, 𝑐}) = 3)) |
| 48 | | hashprlei 14507 |
. . . . . . . . . . . . . 14
⊢ ({𝑎, 𝑐} ∈ Fin ∧ (♯‘{𝑎, 𝑐}) ≤ 2) |
| 49 | | breq1 5146 |
. . . . . . . . . . . . . . . . 17
⊢
((♯‘{𝑎,
𝑐}) = 3 →
((♯‘{𝑎, 𝑐}) ≤ 2 ↔ 3 ≤
2)) |
| 50 | 49, 21 | biimtrdi 253 |
. . . . . . . . . . . . . . . 16
⊢
((♯‘{𝑎,
𝑐}) = 3 →
((♯‘{𝑎, 𝑐}) ≤ 2 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 51 | 50 | com12 32 |
. . . . . . . . . . . . . . 15
⊢
((♯‘{𝑎,
𝑐}) ≤ 2 →
((♯‘{𝑎, 𝑐}) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 52 | 51 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (({𝑎, 𝑐} ∈ Fin ∧ (♯‘{𝑎, 𝑐}) ≤ 2) → ((♯‘{𝑎, 𝑐}) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 53 | 48, 52 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
((♯‘{𝑎,
𝑐}) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
| 54 | 47, 53 | biimtrdi 253 |
. . . . . . . . . . . 12
⊢ (𝑉 = {𝑎, 𝑐} → ((♯‘𝑉) = 3 → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 55 | 54 | adantld 490 |
. . . . . . . . . . 11
⊢ (𝑉 = {𝑎, 𝑐} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 56 | 46, 55 | biimtrdi 253 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑐 → (𝑉 = {𝑎, 𝑏, 𝑐} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))) |
| 57 | 28, 42, 56 | 3jaoi 1430 |
. . . . . . . . 9
⊢ ((𝑎 = 𝑏 ∨ 𝑎 = 𝑐 ∨ 𝑏 = 𝑐) → (𝑉 = {𝑎, 𝑏, 𝑐} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)))) |
| 58 | 57 | impcomd 411 |
. . . . . . . 8
⊢ ((𝑎 = 𝑏 ∨ 𝑎 = 𝑐 ∨ 𝑏 = 𝑐) → (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 59 | 8, 58 | sylbi 217 |
. . . . . . 7
⊢ (¬
(𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) → (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
| 60 | 2, 59 | pm2.61i 182 |
. . . . . 6
⊢ (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}) → (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
| 61 | | simpr 484 |
. . . . . 6
⊢ (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}) → 𝑉 = {𝑎, 𝑏, 𝑐}) |
| 62 | 60, 61 | jca 511 |
. . . . 5
⊢ (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}) → ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐})) |
| 63 | 62 | ex 412 |
. . . 4
⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (𝑉 = {𝑎, 𝑏, 𝑐} → ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}))) |
| 64 | 63 | eximdv 1917 |
. . 3
⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐} → ∃𝑐((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}))) |
| 65 | 64 | 2eximdv 1919 |
. 2
⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → (∃𝑎∃𝑏∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐} → ∃𝑎∃𝑏∃𝑐((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}))) |
| 66 | 1, 65 | mpd 15 |
1
⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐})) |