Step |
Hyp |
Ref |
Expression |
1 |
|
pltnlt.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pltnlt.s |
⊢ < = ( lt ‘ 𝐾 ) |
3 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
4 |
1 3 2
|
pltnle |
⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ¬ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) |
5 |
4
|
ex |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ¬ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ) |
6 |
3 2
|
pltle |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 < 𝑋 → 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ) |
7 |
6
|
3com23 |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 < 𝑋 → 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ) |
8 |
5 7
|
nsyld |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ¬ 𝑌 < 𝑋 ) ) |
9 |
|
imnan |
⊢ ( ( 𝑋 < 𝑌 → ¬ 𝑌 < 𝑋 ) ↔ ¬ ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑋 ) ) |
10 |
8 9
|
sylib |
⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ¬ ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑋 ) ) |