Step |
Hyp |
Ref |
Expression |
1 |
|
1psubcl.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
2 |
|
1psubcl.c |
⊢ 𝐶 = ( PSubCl ‘ 𝐾 ) |
3 |
|
snssi |
⊢ ( 𝑄 ∈ 𝐴 → { 𝑄 } ⊆ 𝐴 ) |
4 |
3
|
adantl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → { 𝑄 } ⊆ 𝐴 ) |
5 |
|
eqid |
⊢ ( ⊥𝑃 ‘ 𝐾 ) = ( ⊥𝑃 ‘ 𝐾 ) |
6 |
1 5
|
2polatN |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ { 𝑄 } ) ) = { 𝑄 } ) |
7 |
1 5 2
|
ispsubclN |
⊢ ( 𝐾 ∈ HL → ( { 𝑄 } ∈ 𝐶 ↔ ( { 𝑄 } ⊆ 𝐴 ∧ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ { 𝑄 } ) ) = { 𝑄 } ) ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → ( { 𝑄 } ∈ 𝐶 ↔ ( { 𝑄 } ⊆ 𝐴 ∧ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ { 𝑄 } ) ) = { 𝑄 } ) ) ) |
9 |
4 6 8
|
mpbir2and |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → { 𝑄 } ∈ 𝐶 ) |