Description: The intersection of two sets is finite if one of them is. (Contributed by Thierry Arnoux, 14-Feb-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | infi | ⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∩ 𝐵 ) ∈ Fin ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 | ⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 | |
2 | ssfi | ⊢ ( ( 𝐴 ∈ Fin ∧ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ) → ( 𝐴 ∩ 𝐵 ) ∈ Fin ) | |
3 | 1 2 | mpan2 | ⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∩ 𝐵 ) ∈ Fin ) |