Theorem elpredimg

Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 13-Apr-2011) (Revised by NM, 5-Apr-2016) (Proof shortened by BJ, 16-Oct-2024)

	Ref	Expression
Assertion	elpredimg	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) → 𝑌 𝑅 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		elpredgg	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) → ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 𝑅 𝑋 ) ) )
2		simpr	⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝑌 𝑅 𝑋 ) → 𝑌 𝑅 𝑋 )
3	1 2	biimtrdi	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) → ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) → 𝑌 𝑅 𝑋 ) )
4	3	syldbl2	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) → 𝑌 𝑅 𝑋 )