Step |
Hyp |
Ref |
Expression |
1 |
|
atnlt.s |
⊢ < = ( lt ‘ 𝐾 ) |
2 |
|
atnlt.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
1
|
pltirr |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → ¬ 𝑃 < 𝑃 ) |
4 |
3
|
3adant3 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑃 < 𝑃 ) |
5 |
|
breq2 |
⊢ ( 𝑃 = 𝑄 → ( 𝑃 < 𝑃 ↔ 𝑃 < 𝑄 ) ) |
6 |
5
|
notbid |
⊢ ( 𝑃 = 𝑄 → ( ¬ 𝑃 < 𝑃 ↔ ¬ 𝑃 < 𝑄 ) ) |
7 |
4 6
|
syl5ibcom |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 = 𝑄 → ¬ 𝑃 < 𝑄 ) ) |
8 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
9 |
8 1
|
pltle |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 < 𝑄 → 𝑃 ( le ‘ 𝐾 ) 𝑄 ) ) |
10 |
8 2
|
atcmp |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ( le ‘ 𝐾 ) 𝑄 ↔ 𝑃 = 𝑄 ) ) |
11 |
9 10
|
sylibd |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 < 𝑄 → 𝑃 = 𝑄 ) ) |
12 |
11
|
necon3ad |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 → ¬ 𝑃 < 𝑄 ) ) |
13 |
7 12
|
pm2.61dne |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑃 < 𝑄 ) |