Description: The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011)
Ref | Expression | ||
---|---|---|---|
Hypotheses | cvrle.b | ⊢ 𝐵 = ( Base ‘ 𝐾 ) | |
cvrle.l | ⊢ ≤ = ( le ‘ 𝐾 ) | ||
cvrle.c | ⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) | ||
Assertion | cvrnle | ⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ¬ 𝑌 ≤ 𝑋 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrle.b | ⊢ 𝐵 = ( Base ‘ 𝐾 ) | |
2 | cvrle.l | ⊢ ≤ = ( le ‘ 𝐾 ) | |
3 | cvrle.c | ⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) | |
4 | eqid | ⊢ ( lt ‘ 𝐾 ) = ( lt ‘ 𝐾 ) | |
5 | 1 4 3 | cvrlt | ⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 ( lt ‘ 𝐾 ) 𝑌 ) |
6 | 1 2 4 | pltnle | ⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ( lt ‘ 𝐾 ) 𝑌 ) → ¬ 𝑌 ≤ 𝑋 ) |
7 | 5 6 | syldan | ⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ¬ 𝑌 ≤ 𝑋 ) |