Step |
Hyp |
Ref |
Expression |
1 |
|
pleval2.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pleval2.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
pleval2.s |
⊢ < = ( lt ‘ 𝐾 ) |
4 |
1 2 3
|
pleval2i |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) ) ) |
5 |
4
|
3adant1 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) ) ) |
6 |
2 3
|
pltle |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → 𝑋 ≤ 𝑌 ) ) |
7 |
1 2
|
posref |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ≤ 𝑋 ) |
8 |
7
|
3adant3 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ≤ 𝑋 ) |
9 |
|
breq2 |
⊢ ( 𝑋 = 𝑌 → ( 𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌 ) ) |
10 |
8 9
|
syl5ibcom |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = 𝑌 → 𝑋 ≤ 𝑌 ) ) |
11 |
6 10
|
jaod |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) → 𝑋 ≤ 𝑌 ) ) |
12 |
5 11
|
impbid |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) ) ) |