Step |
Hyp |
Ref |
Expression |
1 |
|
hlsupr.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
hlsupr.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
hlsupr.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
5 |
4 1 2 3
|
hlsuprexch |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝑃 ≠ 𝑄 → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ∀ 𝑟 ∈ ( Base ‘ 𝐾 ) ( ( ¬ 𝑃 ≤ 𝑟 ∧ 𝑃 ≤ ( 𝑟 ∨ 𝑄 ) ) → 𝑄 ≤ ( 𝑟 ∨ 𝑃 ) ) ) ) |
6 |
5
|
simpld |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) |
7 |
6
|
imp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |