MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpeq0 Structured version   Unicode version

Theorem xpeq0 5277
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 5276 . . 3  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( A  X.  B )  =/=  (/) )
21necon2bbii 2689 . 2  |-  ( ( A  X.  B )  =  (/)  <->  -.  ( A  =/=  (/)  /\  B  =/=  (/) ) )
3 ianor 488 . 2  |-  ( -.  ( A  =/=  (/)  /\  B  =/=  (/) )  <->  ( -.  A  =/=  (/)  \/  -.  B  =/=  (/) ) )
4 nne 2626 . . 3  |-  ( -.  A  =/=  (/)  <->  A  =  (/) )
5 nne 2626 . . 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 1369    =/= wne 2620   (/)c0 3656    X. cxp 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-br 4312  df-opab 4370  df-xp 4865  df-rel 4866  df-cnv 4867
This theorem is referenced by:  xpcan  5293  xpcan2  5294  frxp  6701  rankxplim3  8107  xpcbas  15007  metn0  19954  filnetlem4  28625
  Copyright terms: Public domain W3C validator