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Theorem 0nelxp 4398
Description: The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
0nelxp  |-  -.  (/)  e.  ( A  X.  B )

Proof of Theorem 0nelxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2605 . . . . . 6  |-  x  e. 
_V
2 vex 2605 . . . . . 6  |-  y  e. 
_V
31, 2opnzi 3998 . . . . 5  |-  <. x ,  y >.  =/=  (/)
4 simpl 107 . . . . . . 7  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  (/)  =  <. x ,  y >. )
54eqcomd 2087 . . . . . 6  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  <. x ,  y >.  =  (/) )
65necon3ai 2295 . . . . 5  |-  ( <.
x ,  y >.  =/=  (/)  ->  -.  ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) ) )
73, 6ax-mp 7 . . . 4  |-  -.  ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )
87nex 1430 . . 3  |-  -.  E. y ( (/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)
98nex 1430 . 2  |-  -.  E. x E. y ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )
10 elxp 4388 . 2  |-  ( (/)  e.  ( A  X.  B
)  <->  E. x E. y
( (/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
) )
119, 10mtbir 629 1  |-  -.  (/)  e.  ( A  X.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434    =/= wne 2246   (/)c0 3258   <.cop 3409    X. cxp 4369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848  df-xp 4377
This theorem is referenced by:  dmsn0  4818  nfunv  4963  reldmtpos  5902  dmtpos  5905  0ncn  7062
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