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 ) ) |
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 ) ) |