Step |
Hyp |
Ref |
Expression |
1 |
|
ople1.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
ople1.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
ople1.u |
⊢ 1 = ( 1. ‘ 𝐾 ) |
4 |
1 2 3
|
ople1 |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ≤ 1 ) |
5 |
4
|
biantrurd |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( 1 ≤ 𝑋 ↔ ( 𝑋 ≤ 1 ∧ 1 ≤ 𝑋 ) ) ) |
6 |
|
opposet |
⊢ ( 𝐾 ∈ OP → 𝐾 ∈ Poset ) |
7 |
6
|
adantr |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ Poset ) |
8 |
|
simpr |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
9 |
1 3
|
op1cl |
⊢ ( 𝐾 ∈ OP → 1 ∈ 𝐵 ) |
10 |
9
|
adantr |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → 1 ∈ 𝐵 ) |
11 |
1 2
|
posasymb |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵 ) → ( ( 𝑋 ≤ 1 ∧ 1 ≤ 𝑋 ) ↔ 𝑋 = 1 ) ) |
12 |
7 8 10 11
|
syl3anc |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 ≤ 1 ∧ 1 ≤ 𝑋 ) ↔ 𝑋 = 1 ) ) |
13 |
5 12
|
bitrd |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( 1 ≤ 𝑋 ↔ 𝑋 = 1 ) ) |