Theorem unixp0 5669

Description: A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)

Assertion

Ref	Expression
unixp0	|- ( ( A X. B ) = (/) <-> U. ( A X. B ) = (/) )

Proof of Theorem unixp0

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4444	. . 3 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = U. (/) )
2		uni0 4465	. . 3 |- U. (/) = (/)
3	1, 2	syl6eq 2672	. 2 |- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) )
4		n0 3931	. . . 4 |- ( ( A X. B ) =/= (/) <-> E. z z e. ( A X. B ) )
5		elxp3 5169	. . . . . 6 |- ( z e. ( A X. B ) <-> E. x E. y ( <. x , y >. = z /\ <. x , y >. e. ( A X. B ) ) )
6		elssuni 4467	. . . . . . . . 9 |- ( <. x , y >. e. ( A X. B ) -> <. x , y >. C_ U. ( A X. B ) )
7		vex 3203	. . . . . . . . . 10 |- x e. _V
8		vex 3203	. . . . . . . . . 10 |- y e. _V
9	7, 8	opnzi 4943	. . . . . . . . 9 |- <. x , y >. =/= (/)
10		ssn0 3976	. . . . . . . . 9 |- ( ( <. x , y >. C_ U. ( A X. B ) /\ <. x , y >. =/= (/) ) -> U. ( A X. B ) =/= (/) )
11	6, 9, 10	sylancl 694	. . . . . . . 8 |- ( <. x , y >. e. ( A X. B ) -> U. ( A X. B ) =/= (/) )
12	11	adantl 482	. . . . . . 7 |- ( ( <. x , y >. = z /\ <. x , y >. e. ( A X. B ) ) -> U. ( A X. B ) =/= (/) )
13	12	exlimivv 1860	. . . . . 6 |- ( E. x E. y ( <. x , y >. = z /\ <. x , y >. e. ( A X. B ) ) -> U. ( A X. B ) =/= (/) )
14	5, 13	sylbi 207	. . . . 5 |- ( z e. ( A X. B ) -> U. ( A X. B ) =/= (/) )
15	14	exlimiv 1858	. . . 4 |- ( E. z z e. ( A X. B ) -> U. ( A X. B ) =/= (/) )
16	4, 15	sylbi 207	. . 3 |- ( ( A X. B ) =/= (/) -> U. ( A X. B ) =/= (/) )
17	16	necon4i 2829	. 2 |- ( U. ( A X. B ) = (/) -> ( A X. B ) = (/) )
18	3, 17	impbii 199	1 |- ( ( A X. B ) = (/) <-> U. ( A X. B ) = (/) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   <.cop 4183   U.cuni 4436   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ne 2795  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-opab 4713  df-xp 5120

This theorem is referenced by:  rankxpsuc  8745