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Theorem 3xpexg 7733
Description: The Cartesian product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
Assertion
Ref Expression
3xpexg (𝑉𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)

Proof of Theorem 3xpexg
StepHypRef Expression
1 xpexg 7731 . . 3 ((𝑉𝑊𝑉𝑊) → (𝑉 × 𝑉) ∈ V)
21anidms 566 . 2 (𝑉𝑊 → (𝑉 × 𝑉) ∈ V)
3 xpexg 7731 . 2 (((𝑉 × 𝑉) ∈ V ∧ 𝑉𝑊) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
42, 3mpancom 685 1 (𝑉𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3466   × cxp 5665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-opab 5202  df-xp 5673  df-rel 5674
This theorem is referenced by: (None)
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