Step |
Hyp |
Ref |
Expression |
1 |
|
llnset.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
llnset.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
3 |
|
llnset.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
llnset.n |
⊢ 𝑁 = ( LLines ‘ 𝐾 ) |
5 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋 ∈ 𝐵 ) |
6 |
|
breq1 |
⊢ ( 𝑝 = 𝑃 → ( 𝑝 𝐶 𝑋 ↔ 𝑃 𝐶 𝑋 ) ) |
7 |
6
|
rspcev |
⊢ ( ( 𝑃 ∈ 𝐴 ∧ 𝑃 𝐶 𝑋 ) → ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) |
8 |
7
|
3ad2antl3 |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) |
9 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝐾 ∈ 𝐷 ) |
10 |
1 2 3 4
|
islln |
⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) ) |
11 |
9 10
|
syl |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) ) |
12 |
5 8 11
|
mpbir2and |
⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋 ∈ 𝑁 ) |