Theorem elpredim

Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012) (Proof shortened by BJ, 16-Oct-2024)

	Ref	Expression
Hypothesis	elpredim.1	⊢ 𝑋 ∈ V
Assertion	elpredim	⊢ ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) → 𝑌 𝑅 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		elpredim.1	⊢ 𝑋 ∈ V
2		elpredimg	⊢ ( ( 𝑋 ∈ V ∧ 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ) → 𝑌 𝑅 𝑋 )
3	1 2	mpan	⊢ ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) → 𝑌 𝑅 𝑋 )