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Theorem xp0 5114
Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
Assertion
Ref Expression
xp0  |-  ( A  X.  (/) )  =  (/)

Proof of Theorem xp0
StepHypRef Expression
1 xp0r 4784 . . 3  |-  ( (/)  X.  A )  =  (/)
21cnveqi 4872 . 2  |-  `' (
(/)  X.  A )  =  `' (/)
3 cnvxp 5113 . 2  |-  `' (
(/)  X.  A )  =  ( A  X.  (/) )
4 cnv0 5100 . 2  |-  `' (/)  =  (/)
52, 3, 43eqtr3i 2324 1  |-  ( A  X.  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1632   (/)c0 3468    X. cxp 4703   `'ccnv 4704
This theorem is referenced by:  xpnz  5115  xpdisj2  5118  dmxpss  5123  rnxpid  5125  xpcan  5128  unixp  5221  fconst5  5747  difxp1  6170  dfac5lem3  7768  xpcdaen  7825  fpwwe2lem13  8280  comfffval  13617  fuchom  13851  xpccofval  13972  frmdplusg  14492  mulgfval  14584  mulgfvi  14587  ga0  14768  symgplusg  14792  efgval  15042  psrplusg  16142  psrvscafval  16151  opsrle  16233  ply1plusgfvi  16336  txindislem  17343  txhaus  17357  0met  17946  zrdivrng  21115  mbfmcst  23579  0rrv  23669  dfpo2  24183  fixpc  25197  isbnd3  26611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713
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