Theorem pstr

Description: A poset is transitive. (Contributed by NM, 12-May-2008) (Revised by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Assertion	pstr	⊢ ( ( 𝑅 ∈ PosetRel ∧ 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		pslem	⊢ ( 𝑅 ∈ PosetRel → ( ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 ) ∧ ( 𝐴 ∈ ∪ ∪ 𝑅 → 𝐴 𝑅 𝐴 ) ∧ ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐴 ) → 𝐴 = 𝐵 ) ) )
2	1	simp1d	⊢ ( 𝑅 ∈ PosetRel → ( ( 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 ) )
3	2	3impib	⊢ ( ( 𝑅 ∈ PosetRel ∧ 𝐴 𝑅 𝐵 ∧ 𝐵 𝑅 𝐶 ) → 𝐴 𝑅 𝐶 )