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Theorem xp0r 4889
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 4828 . . 3  |-  ( z  e.  ( (/)  X.  A
)  <->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) ) )
2 noel 3568 . . . . . . 7  |-  -.  x  e.  (/)
3 simprl 733 . . . . . . 7  |-  ( ( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) )  ->  x  e.  (/) )
42, 3mto 169 . . . . . 6  |-  -.  (
z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) )
54nex 1561 . . . . 5  |-  -.  E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  (/)  /\  y  e.  A ) )
65nex 1561 . . . 4  |-  -.  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  (/)  /\  y  e.  A
) )
7 noel 3568 . . . 4  |-  -.  z  e.  (/)
86, 72false 340 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) )  <->  z  e.  (/) )
91, 8bitri 241 . 2  |-  ( z  e.  ( (/)  X.  A
)  <->  z  e.  (/) )
109eqriv 2377 1  |-  ( (/)  X.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   (/)c0 3564   <.cop 3753    X. cxp 4809
This theorem is referenced by:  dmxpid  5022  res0  5083  xp0  5224  xpnz  5225  xpdisj1  5227  xpcan2  5239  xpima  5246  unixp  5335  unixpid  5337  xpcoid  5348  difxp2  6314  fodomr  7187  xpfi  7307  cdaassen  7988  iundom2g  8341  alephadd  8378  hashxplem  11616  ramcl  13317  txindislem  17579  txhaus  17593  tmdgsum  18039  ust0  18163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-opab 4201  df-xp 4817
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