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Theorem 3xpexg 6584
Description: The Cartesian product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
Assertion
Ref Expression
3xpexg  |-  ( V  e.  W  ->  (
( V  X.  V
)  X.  V )  e.  _V )

Proof of Theorem 3xpexg
StepHypRef Expression
1 xpexg 6583 . . 3  |-  ( ( V  e.  W  /\  V  e.  W )  ->  ( V  X.  V
)  e.  _V )
21anidms 645 . 2  |-  ( V  e.  W  ->  ( V  X.  V )  e. 
_V )
3 xpexg 6583 . 2  |-  ( ( ( V  X.  V
)  e.  _V  /\  V  e.  W )  ->  ( ( V  X.  V )  X.  V
)  e.  _V )
42, 3mpancom 669 1  |-  ( V  e.  W  ->  (
( V  X.  V
)  X.  V )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1802   _Vcvv 3093    X. cxp 4983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rex 2797  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-opab 4492  df-xp 4991
This theorem is referenced by:  2wlkonot  24730  2spthonot  24731  2wlksot  24732  2spthsot  24733  usg2spot2nb  24930  usgreg2spot  24932  2spotmdisj  24933
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