Theorem altxpexg 32085

Description: The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)

Assertion

Ref	Expression
altxpexg	|- ( ( A e. V /\ B e. W ) -> ( A XX. B ) e. _V )

Proof of Theorem altxpexg

Step	Hyp	Ref	Expression
1		altxpsspw 32084	. 2 |- ( A XX. B ) C_ ~P ~P ( A u. ~P B )
2		pwexg 4850	. . . 4 |- ( B e. W -> ~P B e. _V )
3		unexg 6959	. . . 4 |- ( ( A e. V /\ ~P B e. _V ) -> ( A u. ~P B ) e. _V )
4	2, 3	sylan2 491	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A u. ~P B ) e. _V )
5		pwexg 4850	. . 3 |- ( ( A u. ~P B ) e. _V -> ~P ( A u. ~P B ) e. _V )
6		pwexg 4850	. . 3 |- ( ~P ( A u. ~P B ) e. _V -> ~P ~P ( A u. ~P B ) e. _V )
7	4, 5, 6	3syl 18	. 2 |- ( ( A e. V /\ B e. W ) -> ~P ~P ( A u. ~P B ) e. _V )
8		ssexg 4804	. 2 |- ( ( ( A XX. B ) C_ ~P ~P ( A u. ~P B ) /\ ~P ~P ( A u. ~P B ) e. _V ) -> ( A XX. B ) e. _V )
9	1, 7, 8	sylancr 695	1 |- ( ( A e. V /\ B e. W ) -> ( A XX. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   ~Pcpw 4158   XX. caltxp 32064

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-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-altop 32065  df-altxp 32066

This theorem is referenced by: (None)