Theorem 3xpexg

Description: The Cartesian product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018)

	Ref	Expression
Assertion	3xpexg	⊢ ( 𝑉 ∈ 𝑊 → ( ( 𝑉 × 𝑉 ) × 𝑉 ) ∈ V )

Proof

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