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Theorem 3xpexg 7464
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 7462 . . 3 ((𝑉𝑊𝑉𝑊) → (𝑉 × 𝑉) ∈ V)
21anidms 567 . 2 (𝑉𝑊 → (𝑉 × 𝑉) ∈ V)
3 xpexg 7462 . 2 (((𝑉 × 𝑉) ∈ V ∧ 𝑉𝑊) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
42, 3mpancom 684 1 (𝑉𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  Vcvv 3492   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by: (None)
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