Theorem elfpw

Description: Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015) (Revised by Mario Carneiro, 22-Aug-2015)

	Ref	Expression
Assertion	elfpw	⊢ ( 𝐴 ∈ ( 𝒫 𝐵 ∩ Fin ) ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ Fin ) )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( 𝐴 ∈ ( 𝒫 𝐵 ∩ Fin ) ↔ ( 𝐴 ∈ 𝒫 𝐵 ∧ 𝐴 ∈ Fin ) )
2		elpwg	⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )
3	2	pm5.32ri	⊢ ( ( 𝐴 ∈ 𝒫 𝐵 ∧ 𝐴 ∈ Fin ) ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ Fin ) )
4	1 3	bitri	⊢ ( 𝐴 ∈ ( 𝒫 𝐵 ∩ Fin ) ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ∈ Fin ) )