Step |
Hyp |
Ref |
Expression |
1 |
|
pltletr.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pltletr.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
pltletr.s |
⊢ < = ( lt ‘ 𝐾 ) |
4 |
1 2 3
|
pleval2 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) ) ) |
5 |
4
|
3adant3r3 |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) ) ) |
6 |
1 3
|
plttr |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑍 ) → 𝑋 < 𝑍 ) ) |
7 |
6
|
expd |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 < 𝑌 → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) ) ) |
8 |
|
breq1 |
⊢ ( 𝑋 = 𝑌 → ( 𝑋 < 𝑍 ↔ 𝑌 < 𝑍 ) ) |
9 |
8
|
biimprd |
⊢ ( 𝑋 = 𝑌 → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) ) |
10 |
9
|
a1i |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 = 𝑌 → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) ) ) |
11 |
7 10
|
jaod |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) ) ) |
12 |
5 11
|
sylbid |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) ) ) |
13 |
12
|
impd |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 < 𝑍 ) → 𝑋 < 𝑍 ) ) |