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Theorem 0nelxp 5036
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 3112 . . . . . 6  |-  x  e. 
_V
2 vex 3112 . . . . . 6  |-  y  e. 
_V
31, 2opnzi 4728 . . . . 5  |-  <. x ,  y >.  =/=  (/)
4 simpl 457 . . . . . . 7  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  (/)  =  <. x ,  y >. )
54eqcomd 2465 . . . . . 6  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  <. x ,  y >.  =  (/) )
65necon3ai 2685 . . . . 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 1628 . . 3  |-  -.  E. y ( (/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)
98nex 1628 . 2  |-  -.  E. x E. y ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )
10 elxp 5025 . 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 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   (/)c0 3793   <.cop 4038    X. cxp 5006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516  df-xp 5014
This theorem is referenced by:  onxpdisj  5092  dmsn0  5481  nfunv  5625  mpt2xopx0ov0  6962  reldmtpos  6981  dmtpos  6985  0nnq  9319  adderpq  9351  mulerpq  9352  lterpq  9365  0ncn  9527  structcnvcnv  14654  msrrcl  29078
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