Theorem elpredgg

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011) Generalize to closed form. (Revised by BJ, 16-Oct-2024)

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

Proof

Step	Hyp	Ref	Expression
1		df-pred	⊢ Pred ( 𝑅 , 𝐴 , 𝑋 ) = ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) )
2	1	elin2	⊢ ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 ∈ ( ◡ 𝑅 “ { 𝑋 } ) ) )
3		elinisegg	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → ( 𝑌 ∈ ( ◡ 𝑅 “ { 𝑋 } ) ↔ 𝑌 𝑅 𝑋 ) )
4	3	anbi2d	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → ( ( 𝑌 ∈ 𝐴 ∧ 𝑌 ∈ ( ◡ 𝑅 “ { 𝑋 } ) ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 𝑅 𝑋 ) ) )
5	2 4	syl5bb	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ) → ( 𝑌 ∈ Pred ( 𝑅 , 𝐴 , 𝑋 ) ↔ ( 𝑌 ∈ 𝐴 ∧ 𝑌 𝑅 𝑋 ) ) )