Step |
Hyp |
Ref |
Expression |
1 |
|
dalema.ph |
⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) ) |
2 |
|
dalemc.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dalemc.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dalemc.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dalem18.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
6 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
7 |
1
|
dalempea |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
8 |
1
|
dalemqea |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
9 |
1
|
dalemrea |
⊢ ( 𝜑 → 𝑅 ∈ 𝐴 ) |
10 |
3 2 4
|
3dim3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
11 |
6 7 8 9 10
|
syl13anc |
⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
12 |
5
|
breq2i |
⊢ ( 𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
13 |
12
|
notbii |
⊢ ( ¬ 𝑐 ≤ 𝑌 ↔ ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
14 |
13
|
rexbii |
⊢ ( ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌 ↔ ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
15 |
11 14
|
sylibr |
⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌 ) |