Step |
Hyp |
Ref |
Expression |
1 |
|
hlatle.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
hlatle.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
hlatle.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
ralbiim |
⊢ ( ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 ↔ 𝑝 ≤ 𝑌 ) ↔ ( ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌 ) ∧ ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋 ) ) ) |
5 |
1 2 3
|
hlatle |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌 ) ) ) |
6 |
1 2 3
|
hlatle |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 ≤ 𝑋 ↔ ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋 ) ) ) |
7 |
6
|
3com23 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 ≤ 𝑋 ↔ ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋 ) ) ) |
8 |
5 7
|
anbi12d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ ( ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 → 𝑝 ≤ 𝑌 ) ∧ ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋 ) ) ) ) |
9 |
4 8
|
bitr4id |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 ↔ 𝑝 ≤ 𝑌 ) ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ) ) |
10 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
11 |
1 2
|
latasymb |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ 𝑋 = 𝑌 ) ) |
12 |
10 11
|
syl3an1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ 𝑋 = 𝑌 ) ) |
13 |
9 12
|
bitrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ 𝑋 ↔ 𝑝 ≤ 𝑌 ) ↔ 𝑋 = 𝑌 ) ) |