Metamath Proof Explorer
Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996)
|
|
Ref |
Expression |
|
Assertion |
so2nr |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ¬ ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
sopo |
⊢ ( 𝑅 Or 𝐴 → 𝑅 Po 𝐴 ) |
2 |
|
po2nr |
⊢ ( ( 𝑅 Po 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ¬ ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐵 ) ) |
3 |
1 2
|
sylan |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ¬ ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐵 ) ) |