Step |
Hyp |
Ref |
Expression |
1 |
|
lplnnelln.n |
⊢ 𝑁 = ( LLines ‘ 𝐾 ) |
2 |
|
lplnnelln.p |
⊢ 𝑃 = ( LPlanes ‘ 𝐾 ) |
3 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
5 |
4 2
|
lplnbase |
⊢ ( 𝑋 ∈ 𝑃 → 𝑋 ∈ ( Base ‘ 𝐾 ) ) |
6 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
7 |
4 6
|
latref |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 ) |
8 |
3 5 7
|
syl2an |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 ) |
9 |
6 1 2
|
lplnnlelln |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑋 ∈ 𝑁 ) → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) |
10 |
9
|
3expia |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ) → ( 𝑋 ∈ 𝑁 → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) ) |
11 |
8 10
|
mt2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ) → ¬ 𝑋 ∈ 𝑁 ) |