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Theorem 3xpexg 6598
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 6597 . . 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 6597 . 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 1767   _Vcvv 3118    X. cxp 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rex 2823  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-opab 4512  df-xp 5011
This theorem is referenced by:  2wlksot  24690  2spthsot  24691  usg2spot2nb  24889  usgreg2spot  24891
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