Proof of Theorem 3dim1lem5
| Step | Hyp | Ref
| Expression |
| 1 | | neeq2 3004 |
. . 3
⊢ (𝑞 = 𝑢 → (𝑃 ≠ 𝑞 ↔ 𝑃 ≠ 𝑢)) |
| 2 | | oveq2 7439 |
. . . . 5
⊢ (𝑞 = 𝑢 → (𝑃 ∨ 𝑞) = (𝑃 ∨ 𝑢)) |
| 3 | 2 | breq2d 5155 |
. . . 4
⊢ (𝑞 = 𝑢 → (𝑟 ≤ (𝑃 ∨ 𝑞) ↔ 𝑟 ≤ (𝑃 ∨ 𝑢))) |
| 4 | 3 | notbid 318 |
. . 3
⊢ (𝑞 = 𝑢 → (¬ 𝑟 ≤ (𝑃 ∨ 𝑞) ↔ ¬ 𝑟 ≤ (𝑃 ∨ 𝑢))) |
| 5 | 2 | oveq1d 7446 |
. . . . 5
⊢ (𝑞 = 𝑢 → ((𝑃 ∨ 𝑞) ∨ 𝑟) = ((𝑃 ∨ 𝑢) ∨ 𝑟)) |
| 6 | 5 | breq2d 5155 |
. . . 4
⊢ (𝑞 = 𝑢 → (𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟) ↔ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟))) |
| 7 | 6 | notbid 318 |
. . 3
⊢ (𝑞 = 𝑢 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟) ↔ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟))) |
| 8 | 1, 4, 7 | 3anbi123d 1438 |
. 2
⊢ (𝑞 = 𝑢 → ((𝑃 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟)) ↔ (𝑃 ≠ 𝑢 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟)))) |
| 9 | | breq1 5146 |
. . . 4
⊢ (𝑟 = 𝑣 → (𝑟 ≤ (𝑃 ∨ 𝑢) ↔ 𝑣 ≤ (𝑃 ∨ 𝑢))) |
| 10 | 9 | notbid 318 |
. . 3
⊢ (𝑟 = 𝑣 → (¬ 𝑟 ≤ (𝑃 ∨ 𝑢) ↔ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢))) |
| 11 | | oveq2 7439 |
. . . . 5
⊢ (𝑟 = 𝑣 → ((𝑃 ∨ 𝑢) ∨ 𝑟) = ((𝑃 ∨ 𝑢) ∨ 𝑣)) |
| 12 | 11 | breq2d 5155 |
. . . 4
⊢ (𝑟 = 𝑣 → (𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟) ↔ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))) |
| 13 | 12 | notbid 318 |
. . 3
⊢ (𝑟 = 𝑣 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟) ↔ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))) |
| 14 | 10, 13 | 3anbi23d 1441 |
. 2
⊢ (𝑟 = 𝑣 → ((𝑃 ≠ 𝑢 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟)) ↔ (𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣)))) |
| 15 | | breq1 5146 |
. . . 4
⊢ (𝑠 = 𝑤 → (𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣) ↔ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))) |
| 16 | 15 | notbid 318 |
. . 3
⊢ (𝑠 = 𝑤 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣) ↔ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))) |
| 17 | 16 | 3anbi3d 1444 |
. 2
⊢ (𝑠 = 𝑤 → ((𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣)) ↔ (𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣)))) |
| 18 | 8, 14, 17 | rspc3ev 3639 |
1
⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟))) |