Step |
Hyp |
Ref |
Expression |
1 |
|
posi.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
posi.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
simp1 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Poset ) |
4 |
|
simp2 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
5 |
|
simp3 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
6 |
1 2
|
posi |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑋 ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) → 𝑋 = 𝑌 ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌 ) → 𝑋 ≤ 𝑌 ) ) ) |
7 |
3 4 5 5 6
|
syl13anc |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑋 ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) → 𝑋 = 𝑌 ) ∧ ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑌 ) → 𝑋 ≤ 𝑌 ) ) ) |
8 |
7
|
simp2d |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) → 𝑋 = 𝑌 ) ) |
9 |
1 2
|
posref |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ≤ 𝑋 ) |
10 |
|
breq2 |
⊢ ( 𝑋 = 𝑌 → ( 𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌 ) ) |
11 |
9 10
|
syl5ibcom |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 = 𝑌 → 𝑋 ≤ 𝑌 ) ) |
12 |
|
breq1 |
⊢ ( 𝑋 = 𝑌 → ( 𝑋 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋 ) ) |
13 |
9 12
|
syl5ibcom |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 = 𝑌 → 𝑌 ≤ 𝑋 ) ) |
14 |
11 13
|
jcad |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 = 𝑌 → ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ) ) |
15 |
14
|
3adant3 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = 𝑌 → ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ) ) |
16 |
8 15
|
impbid |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ 𝑋 = 𝑌 ) ) |