Theorem 3xpexg 6961

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

Step	Hyp	Ref	Expression
1		xpexg 6960	. . 3 |- ( ( V e. W /\ V e. W ) -> ( V X. V ) e. _V )
2	1	anidms 677	. 2 |- ( V e. W -> ( V X. V ) e. _V )
3		xpexg 6960	. 2 |- ( ( ( V X. V ) e. _V /\ V e. W ) -> ( ( V X. V ) X. V ) e. _V )
4	2, 3	mpancom 703	1 |- ( V e. W -> ( ( V X. V ) X. V ) e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)