Theorem predexg

Description: The predecessor class exists when A does. (Contributed by Scott Fenton, 8-Feb-2011) Generalize to closed form. (Revised by BJ, 27-Oct-2024)

	Ref	Expression
Assertion	predexg	⊢ ( 𝐴 ∈ 𝑉 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		df-pred	⊢ Pred ( 𝑅 , 𝐴 , 𝑋 ) = ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) )
2		inex1g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑋 } ) ) ∈ V )
3	1 2	eqeltrid	⊢ ( 𝐴 ∈ 𝑉 → Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ V )