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Theorem 3xpexg 30291
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 6620 . . 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 6620 . 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 1758   _Vcvv 3078    X. cxp 4949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rex 2805  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-opab 4462  df-xp 4957
This theorem is referenced by:  2wlksot  30557  2spthsot  30558  usg2spot2nb  30829  usgreg2spot  30831
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