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Theorem 3xpexg 6917
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 6916 . . 3 ((𝑉𝑊𝑉𝑊) → (𝑉 × 𝑉) ∈ V)
21anidms 676 . 2 (𝑉𝑊 → (𝑉 × 𝑉) ∈ V)
3 xpexg 6916 . 2 (((𝑉 × 𝑉) ∈ V ∧ 𝑉𝑊) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
42, 3mpancom 702 1 (𝑉𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Vcvv 3186   × cxp 5074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-opab 4676  df-xp 5082  df-rel 5083
This theorem is referenced by: (None)
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