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Theorem 0nelxp 5026
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 3116 . . . . . 6  |-  x  e. 
_V
2 vex 3116 . . . . . 6  |-  y  e. 
_V
31, 2opnzi 4719 . . . . 5  |-  <. x ,  y >.  =/=  (/)
4 simpl 457 . . . . . . 7  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  (/)  =  <. x ,  y >. )
54eqcomd 2475 . . . . . 6  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  <. x ,  y >.  =  (/) )
65necon3ai 2695 . . . . 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 1610 . . 3  |-  -.  E. y ( (/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)
98nex 1610 . 2  |-  -.  E. x E. y ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )
10 elxp 5016 . 2  |-  ( (/)  e.  ( A  X.  B
)  <->  E. x E. y
( (/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
) )
119, 10mtbir 299 1  |-  -.  (/)  e.  ( A  X.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   (/)c0 3785   <.cop 4033    X. cxp 4997
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 4568  ax-nul 4576  ax-pr 4686
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005
This theorem is referenced by:  onxpdisj  5081  dmsn0  5473  nfunv  5617  mpt2xopx0ov0  6941  reldmtpos  6960  dmtpos  6964  0nnq  9298  adderpq  9330  mulerpq  9331  lterpq  9344  0ncn  9506  structcnvcnv  14494
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