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Theorem 3xpexg 7450
 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 7448 . . 3 ((𝑉𝑊𝑉𝑊) → (𝑉 × 𝑉) ∈ V)
21anidms 570 . 2 (𝑉𝑊 → (𝑉 × 𝑉) ∈ V)
3 xpexg 7448 . 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 2115  Vcvv 3471   × cxp 5526 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-opab 5102  df-xp 5534  df-rel 5535 This theorem is referenced by: (None)
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