MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0nelxp Structured version   Unicode version

Theorem 0nelxp 4908
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 2961 . . . . . 6  |-  x  e. 
_V
2 vex 2961 . . . . . 6  |-  y  e. 
_V
31, 2opnzi 4435 . . . . 5  |-  <. x ,  y >.  =/=  (/)
4 simpl 445 . . . . . . 7  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  (/)  =  <. x ,  y >. )
54eqcomd 2443 . . . . . 6  |-  ( (
(/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  ->  <. x ,  y >.  =  (/) )
65necon3ai 2646 . . . . 5  |-  ( <.
x ,  y >.  =/=  (/)  ->  -.  ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) ) )
73, 6ax-mp 8 . . . 4  |-  -.  ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )
87nex 1565 . . 3  |-  -.  E. y ( (/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)
98nex 1565 . 2  |-  -.  E. x E. y ( (/)  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )
10 elxp 4897 . 2  |-  ( (/)  e.  ( A  X.  B
)  <->  E. x E. y
( (/)  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
) )
119, 10mtbir 292 1  |-  -.  (/)  e.  ( A  X.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   (/)c0 3630   <.cop 3819    X. cxp 4878
This theorem is referenced by:  onxpdisj  4959  dmsn0  5339  nfunv  5486  mpt2xopx0ov0  6469  reldmtpos  6489  dmtpos  6493  0nnq  8803  adderpq  8835  mulerpq  8836  lterpq  8849  0ncn  9010  structcnvcnv  13482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886
  Copyright terms: Public domain W3C validator