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Theorem xp0r 4770
Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
xp0r  |-  ( (/)  X.  A )  =  (/)

Proof of Theorem xp0r
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4708 . . 3  |-  ( z  e.  ( (/)  X.  A
)  <->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) ) )
2 noel 3461 . . . . . . 7  |-  -.  x  e.  (/)
3 simprl 732 . . . . . . 7  |-  ( ( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) )  ->  x  e.  (/) )
42, 3mto 167 . . . . . 6  |-  -.  (
z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) )
54nex 1544 . . . . 5  |-  -.  E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  (/)  /\  y  e.  A ) )
65nex 1544 . . . 4  |-  -.  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  (/)  /\  y  e.  A
) )
7 noel 3461 . . . 4  |-  -.  z  e.  (/)
86, 72false 339 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) )  <->  z  e.  (/) )
91, 8bitri 240 . 2  |-  ( z  e.  ( (/)  X.  A
)  <->  z  e.  (/) )
109eqriv 2282 1  |-  ( (/)  X.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1530    = wceq 1625    e. wcel 1686   (/)c0 3457   <.cop 3645    X. cxp 4689
This theorem is referenced by:  dmxpid  4900  res0  4961  xp0  5100  xpnz  5101  xpdisj1  5103  xpcan2  5115  unixp  5207  unixpid  5209  difxp2  6157  fodomr  7014  xpfi  7130  cdaassen  7810  iundom2g  8164  alephadd  8201  hashxplem  11387  ramcl  13078  txindislem  17329  txhaus  17343  tmdgsum  17780  xpima  23204  fixpc  25105  0alg  25767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-opab 4080  df-xp 4697
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