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Theorem xpeq0 5433
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
StepHypRef Expression
1 xpnz 5432 . . 3  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
21necon2bbii 2734 . 2  |-  ( ( A  X.  B )  =  (/)  <->  -.  ( A  =/=  (/)  /\  B  =/=  (/) ) )
3 ianor 488 . 2  |-  ( -.  ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( -.  A  =/=  (/)  \/  -.  B  =/=  (/) ) )
4 nne 2668 . . 3  |-  ( -.  A  =/=  (/)  <->  A  =  (/) )
5 nne 2668 . . 3  |-  ( -.  B  =/=  (/)  <->  B  =  (/) )
64, 5orbi12i 521 . 2  |-  ( ( -.  A  =/=  (/)  \/  -.  B  =/=  (/) )  <->  ( A  =  (/)  \/  B  =  (/) ) )
72, 3, 63bitri 271 1  |-  ( ( A  X.  B )  =  (/)  <->  ( A  =  (/)  \/  B  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    =/= wne 2662   (/)c0 3790    X. cxp 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013
This theorem is referenced by:  xpcan  5449  xpcan2  5450  frxp  6905  rankxplim3  8311  xpcbas  15321  metn0  20729  filnetlem4  30133
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