Theorem xpeq0 5554

Description: At least one member of an empty Cartesian product is empty. (Contributed by NM, 27-Aug-2006.)

Assertion

Ref	Expression
xpeq0	|- ( ( A X. B ) = (/) <-> ( A = (/) \/ B = (/) ) )

Proof of Theorem xpeq0

Step	Hyp	Ref	Expression
1		xpnz 5553	. . 3 |- ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) )
2	1	necon2bbii 2845	. 2 |- ( ( A X. B ) = (/) <-> -. ( A =/= (/) /\ B =/= (/) ) )
3		ianor 509	. 2 |- ( -. ( A =/= (/) /\ B =/= (/) ) <-> ( -. A =/= (/) \/ -. B =/= (/) ) )
4		nne 2798	. . 3 |- ( -. A =/= (/) <-> A = (/) )
5		nne 2798	. . 3 |- ( -. B =/= (/) <-> B = (/) )
6	4, 5	orbi12i 543	. 2 |- ( ( -. A =/= (/) \/ -. B =/= (/) ) <-> ( A = (/) \/ B = (/) ) )
7	2, 3, 6	3bitri 286	1 |- ( ( A X. B ) = (/) <-> ( A = (/) \/ B = (/) ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   =/= wne 2794   (/)c0 3915   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-eu 2474  df-mo 2475  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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by:  xpcan  5570  xpcan2  5571  frxp  7287  rankxplim3  8744  xpcbas  16818  metn0  22165  filnetlem4  32376