Theorem ssfid

Description: A subset of a finite set is finite, deduction version of ssfi . (Contributed by Glauco Siliprandi, 21-Nov-2020)

	Ref	Expression
Hypotheses	ssfid.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	ssfid.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
Assertion	ssfid	⊢ ( 𝜑 → 𝐵 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		ssfid.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		ssfid.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
3		ssfi	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ∈ Fin )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 𝐵 ∈ Fin )