Theorem elscottab

Description: An element of the output of the Scott operation applied to a class abstraction satisfies the class abstraction's predicate. (Contributed by Rohan Ridenour, 14-Aug-2023)

	Ref	Expression
Hypothesis	elscottab.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	elscottab	⊢ ( 𝑦 ∈ Scott { 𝑥 ∣ 𝜑 } → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		elscottab.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		scottss	⊢ Scott { 𝑥 ∣ 𝜑 } ⊆ { 𝑥 ∣ 𝜑 }
3	2	sseli	⊢ ( 𝑦 ∈ Scott { 𝑥 ∣ 𝜑 } → 𝑦 ∈ { 𝑥 ∣ 𝜑 } )
4		vex	⊢ 𝑦 ∈ V
5	4 1	elab	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 )
6	3 5	sylib	⊢ ( 𝑦 ∈ Scott { 𝑥 ∣ 𝜑 } → 𝜓 )