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Theorem 3xpexg 7754
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 7752 . . 3 ((𝑉𝑊𝑉𝑊) → (𝑉 × 𝑉) ∈ V)
21anidms 566 . 2 (𝑉𝑊 → (𝑉 × 𝑉) ∈ V)
3 xpexg 7752 . 2 (((𝑉 × 𝑉) ∈ V ∧ 𝑉𝑊) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
42, 3mpancom 687 1 (𝑉𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3471   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-opab 5211  df-xp 5684  df-rel 5685
This theorem is referenced by: (None)
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