Theorem isfin1a

Description: Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015)

	Ref	Expression
Assertion	isfin1a	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ FinIa ↔ ∀ 𝑦 ∈ 𝒫 𝐴 ( 𝑦 ∈ Fin ∨ ( 𝐴 ∖ 𝑦 ) ∈ Fin ) ) )

Proof

Step	Hyp	Ref	Expression
1		pweq	⊢ ( 𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴 )
2		difeq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∖ 𝑦 ) = ( 𝐴 ∖ 𝑦 ) )
3	2	eleq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∖ 𝑦 ) ∈ Fin ↔ ( 𝐴 ∖ 𝑦 ) ∈ Fin ) )
4	3	orbi2d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑦 ∈ Fin ∨ ( 𝑥 ∖ 𝑦 ) ∈ Fin ) ↔ ( 𝑦 ∈ Fin ∨ ( 𝐴 ∖ 𝑦 ) ∈ Fin ) ) )
5	1 4	raleqbidv	⊢ ( 𝑥 = 𝐴 → ( ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑦 ∈ Fin ∨ ( 𝑥 ∖ 𝑦 ) ∈ Fin ) ↔ ∀ 𝑦 ∈ 𝒫 𝐴 ( 𝑦 ∈ Fin ∨ ( 𝐴 ∖ 𝑦 ) ∈ Fin ) ) )
6		df-fin1a	⊢ FinIa = { 𝑥 ∣ ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑦 ∈ Fin ∨ ( 𝑥 ∖ 𝑦 ) ∈ Fin ) }
7	5 6	elab2g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ FinIa ↔ ∀ 𝑦 ∈ 𝒫 𝐴 ( 𝑦 ∈ Fin ∨ ( 𝐴 ∖ 𝑦 ) ∈ Fin ) ) )