Theorem potr

Description: A partial order is a transitive relation. (Contributed by NM, 27-Mar-1997)

	Ref	Expression
Assertion	potr	⊢ ( ( 𝑅 Po 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		pocl	⊢ ( 𝑅 Po 𝐴 → ( ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) ) ) )
2	1	imp	⊢ ( ( 𝑅 Po 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) ) )
3	2	simprd	⊢ ( ( 𝑅 Po 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) )