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Theorem xp0r 4949
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 4888 . . 3  |-  ( z  e.  ( (/)  X.  A
)  <->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  (/)  /\  y  e.  A ) ) )
2 noel 3625 . . . . . . 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 1564 . . . . 5  |-  -.  E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  (/)  /\  y  e.  A ) )
65nex 1564 . . . 4  |-  -.  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  (/)  /\  y  e.  A
) )
7 noel 3625 . . . 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 2433 1  |-  ( (/)  X.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   (/)c0 3621   <.cop 3810    X. cxp 4869
This theorem is referenced by:  dmxpid  5082  res0  5143  xp0  5284  xpnz  5285  xpdisj1  5287  xpcan2  5299  xpima  5306  unixp  5395  unixpid  5397  xpcoid  5408  difxp2  6375  fodomr  7251  xpfi  7371  cdaassen  8055  iundom2g  8408  alephadd  8445  hashxplem  11689  ramcl  13390  txindislem  17658  txhaus  17672  tmdgsum  18118  ust0  18242  sibf0  24642  0mbf  26243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-opab 4260  df-xp 4877
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