Step |
Hyp |
Ref |
Expression |
1 |
|
2atneat.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
2atneat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
simpl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝐾 ∈ HL ) |
4 |
|
simpr1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ∈ 𝐴 ) |
5 |
|
simpr2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑄 ∈ 𝐴 ) |
6 |
|
simpr3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ≠ 𝑄 ) |
7 |
|
eqid |
⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 ) |
8 |
1 2 7
|
llni2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( LLines ‘ 𝐾 ) ) |
9 |
3 4 5 6 8
|
syl31anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( LLines ‘ 𝐾 ) ) |
10 |
2 7
|
llnneat |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( LLines ‘ 𝐾 ) ) → ¬ ( 𝑃 ∨ 𝑄 ) ∈ 𝐴 ) |
11 |
9 10
|
syldan |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ¬ ( 𝑃 ∨ 𝑄 ) ∈ 𝐴 ) |