Theorem 3xpexg 7680

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

Assertion

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

Proof of Theorem 3xpexg

Step	Hyp	Ref	Expression
1		xpexg 7678	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑉 ∈ 𝑊) → (𝑉 × 𝑉) ∈ V)
2	1	anidms 566	. 2 ⊢ (𝑉 ∈ 𝑊 → (𝑉 × 𝑉) ∈ V)
3		xpexg 7678	. 2 ⊢ (((𝑉 × 𝑉) ∈ V ∧ 𝑉 ∈ 𝑊) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
4	2, 3	mpancom 688	1 ⊢ (𝑉 ∈ 𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2110  Vcvv 3434   × cxp 5612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-opab 5152  df-xp 5620  df-rel 5621

This theorem is referenced by: (None)