Theorem isfin1-4

Description: A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014) (Revised by Mario Carneiro, 17-May-2015)

	Ref	Expression
Assertion	isfin1-4	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ Fin ↔ [⊊] Fr 𝒫 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		isfin1-3	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ Fin ↔ ◡ [⊊] Fr 𝒫 𝐴 ) )
2		eqid	⊢ ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑥 ) ) = ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑥 ) )
3	2	compssiso	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑥 ) ) Isom [⊊] , ◡ [⊊] ( 𝒫 𝐴 , 𝒫 𝐴 ) )
4		isofr	⊢ ( ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴 ∖ 𝑥 ) ) Isom [⊊] , ◡ [⊊] ( 𝒫 𝐴 , 𝒫 𝐴 ) → ( [⊊] Fr 𝒫 𝐴 ↔ ◡ [⊊] Fr 𝒫 𝐴 ) )
5	3 4	syl	⊢ ( 𝐴 ∈ 𝑉 → ( [⊊] Fr 𝒫 𝐴 ↔ ◡ [⊊] Fr 𝒫 𝐴 ) )
6	1 5	bitr4d	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ Fin ↔ [⊊] Fr 𝒫 𝐴 ) )