Theorem 3xpexg 7731

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 7729	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑉 ∈ 𝑊) → (𝑉 × 𝑉) ∈ V)
2	1	anidms 574	. 2 ⊢ (𝑉 ∈ 𝑊 → (𝑉 × 𝑉) ∈ V)
3		xpexg 7729	. 2 ⊢ (((𝑉 × 𝑉) ∈ V ∧ 𝑉 ∈ 𝑊) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
4	2, 3	mpancom 698	1 ⊢ (𝑉 ∈ 𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 3453   × cxp 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-opab 5162  df-xp 5651  df-rel 5652

This theorem is referenced by: (None)