Theorem 3xpexg 7739

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 7737	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑉 ∈ 𝑊) → (𝑉 × 𝑉) ∈ V)
2	1	anidms 568	. 2 ⊢ (𝑉 ∈ 𝑊 → (𝑉 × 𝑉) ∈ V)
3		xpexg 7737	. 2 ⊢ (((𝑉 × 𝑉) ∈ V ∧ 𝑉 ∈ 𝑊) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
4	2, 3	mpancom 687	1 ⊢ (𝑉 ∈ 𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3475   × cxp 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-opab 5212  df-xp 5683  df-rel 5684

This theorem is referenced by: (None)