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Theorem xp0 5098
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 4768 . . 3  |-  ( (/)  X.  A )  =  (/)
21cnveqi 4856 . 2  |-  `' (
(/)  X.  A )  =  `' (/)
3 cnvxp 5097 . 2  |-  `' (
(/)  X.  A )  =  ( A  X.  (/) )
4 cnv0 5084 . 2  |-  `' (/)  =  (/)
52, 3, 43eqtr3i 2311 1  |-  ( A  X.  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1623   (/)c0 3455    X. cxp 4687   `'ccnv 4688
This theorem is referenced by:  xpnz  5099  xpdisj2  5102  dmxpss  5107  rnxpid  5109  xpcan  5112  unixp  5205  fconst5  5731  difxp1  6154  dfac5lem3  7752  xpcdaen  7809  fpwwe2lem13  8264  comfffval  13601  fuchom  13835  xpccofval  13956  frmdplusg  14476  mulgfval  14568  mulgfvi  14571  ga0  14752  symgplusg  14776  efgval  15026  psrplusg  16126  psrvscafval  16135  opsrle  16217  ply1plusgfvi  16320  txindislem  17327  txhaus  17341  0met  17930  zrdivrng  21099  dfpo2  23523  fixpc  24506  isbnd3  25920
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 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697
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