Theorem 3xpfi

Description: The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018)

	Ref	Expression
Assertion	3xpfi	⊢ ( 𝑉 ∈ Fin → ( ( 𝑉 × 𝑉 ) × 𝑉 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		xpfi	⊢ ( ( 𝑉 ∈ Fin ∧ 𝑉 ∈ Fin ) → ( 𝑉 × 𝑉 ) ∈ Fin )
2	1	anidms	⊢ ( 𝑉 ∈ Fin → ( 𝑉 × 𝑉 ) ∈ Fin )
3		xpfi	⊢ ( ( ( 𝑉 × 𝑉 ) ∈ Fin ∧ 𝑉 ∈ Fin ) → ( ( 𝑉 × 𝑉 ) × 𝑉 ) ∈ Fin )
4	2 3	mpancom	⊢ ( 𝑉 ∈ Fin → ( ( 𝑉 × 𝑉 ) × 𝑉 ) ∈ Fin )