Step |
Hyp |
Ref |
Expression |
1 |
|
leatom.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
leatom.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
leatom.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
4 |
|
leatom.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
1 2 3 4
|
leatb |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑃 ↔ ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ) ) |
6 |
|
orcom |
⊢ ( ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ ( 𝑋 = 0 ∨ 𝑋 = 𝑃 ) ) |
7 |
|
neor |
⊢ ( ( 𝑋 = 0 ∨ 𝑋 = 𝑃 ) ↔ ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) ) |
8 |
6 7
|
bitri |
⊢ ( ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ↔ ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) ) |
9 |
5 8
|
bitrdi |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑃 ↔ ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) ) ) |
10 |
9
|
biimpd |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≤ 𝑃 → ( 𝑋 ≠ 0 → 𝑋 = 𝑃 ) ) ) |
11 |
10
|
com23 |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 ≠ 0 → ( 𝑋 ≤ 𝑃 → 𝑋 = 𝑃 ) ) ) |
12 |
11
|
imp32 |
⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ ( 𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃 ) ) → 𝑋 = 𝑃 ) |