Theorem predso

Description: Property of the predecessor class for strict total orders. (Contributed by Scott Fenton, 11-Feb-2011)

	Ref	Expression
Assertion	predso	⊢ ( ( 𝑅 Or 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) → Pred ( 𝑅 , 𝐴 , 𝑌 ) ⊆ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sopo	⊢ ( 𝑅 Or 𝐴 → 𝑅 Po 𝐴 )
2		predpo	⊢ ( ( 𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) → Pred ( 𝑅 , 𝐴 , 𝑌 ) ⊆ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) )
3	1 2	sylan	⊢ ( ( 𝑅 Or 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) → Pred ( 𝑅 , 𝐴 , 𝑌 ) ⊆ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) )