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Theorem 0nelxp 4867
Description: The empty set is not a member of a Cartesian 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 3034 . . . . . 6  |-  x  e. 
_V
2 vex 3034 . . . . . 6  |-  y  e. 
_V
31, 2opnzi 4674 . . . . 5  |-  <. x ,  y >.  =/=  (/)
4 simpl 464 . . . . . . 7  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  (/)  =  <. x ,  y >. )
54eqcomd 2477 . . . . . 6  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  <. x ,  y >.  =  (/) )
65necon3ai 2668 . . . . 5  |-  ( <.
x ,  y >.  =/=  (/)  ->  -.  ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) ) )
73, 6ax-mp 5 . . . 4  |-  -.  ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )
87nex 1686 . . 3  |-  -.  E. y ( (/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)
98nex 1686 . 2  |-  -.  E. x E. y ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )
10 elxp 4856 . 2  |-  ( (/)  e.  ( A  X.  B
)  <->  E. x E. y
( (/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
) )
119, 10mtbir 306 1  |-  -.  (/)  e.  ( A  X.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   (/)c0 3722   <.cop 3965    X. cxp 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845
This theorem is referenced by:  dmsn0  5310  onxpdisj  5549  nfunv  5620  mpt2xopx0ov0  6981  reldmtpos  6999  dmtpos  7003  0nnq  9367  adderpq  9399  mulerpq  9400  lterpq  9413  0ncn  9575  structcnvcnv  15210  msrrcl  30253  relintabex  36258  vtxval0  39292  iedgval0  39293
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