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Theorem xp0r 4767
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 4705 . . 3  |-  ( z  e.  ( (/)  X.  A
)  <->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) ) )
2 noel 3460 . . . . . . 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 1542 . . . . 5  |-  -.  E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  (/)  /\  y  e.  A ) )
65nex 1542 . . . 4  |-  -.  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  (/)  /\  y  e.  A
) )
7 noel 3460 . . . 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 2281 1  |-  ( (/)  X.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685   (/)c0 3456   <.cop 3644    X. cxp 4686
This theorem is referenced by:  dmxpid  4897  res0  4958  xp0  5097  xpnz  5098  xpdisj1  5100  xpcan2  5112  unixp  5203  unixpid  5205  difxp2  6117  fodomr  7008  xpfi  7124  cdaassen  7804  iundom2g  8158  alephadd  8195  hashxplem  11381  ramcl  13072  txindislem  17323  txhaus  17337  tmdgsum  17774  fixpc  24504  0alg  25167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-opab 4079  df-xp 4694
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