Theorem elpred

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011) (Proof shortened by BJ, 16-Oct-2024)

	Ref	Expression
Hypothesis	elpred.1	⊢ 𝑌 ∈ V
Assertion	elpred	⊢ ( 𝑋 ∈ 𝐷 → ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 𝑅 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elpred.1	⊢ 𝑌 ∈ V
2		elpredgg	⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ V ) → ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 𝑅 𝑋 ) ) )
3	1 2	mpan2	⊢ ( 𝑋 ∈ 𝐷 → ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 𝑅 𝑋 ) ) )