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Theorem xp0 5282
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 4947 . . 3  |-  ( (/)  X.  A )  =  (/)
21cnveqi 5038 . 2  |-  `' (
(/)  X.  A )  =  `' (/)
3 cnvxp 5281 . 2  |-  `' (
(/)  X.  A )  =  ( A  X.  (/) )
4 cnv0 5266 . 2  |-  `' (/)  =  (/)
52, 3, 43eqtr3i 2463 1  |-  ( A  X.  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652   (/)c0 3620    X. cxp 4867   `'ccnv 4868
This theorem is referenced by:  xpnz  5283  xpdisj2  5286  dmxpss  5291  rnxpid  5293  xpcan  5296  unixp  5393  fconst5  5940  difxp1  6372  dfac5lem3  7995  xpcdaen  8052  fpwwe2lem13  8506  comfffval  13912  fuchom  14146  xpccofval  14267  frmdplusg  14787  mulgfval  14879  mulgfvi  14882  ga0  15063  symgplusg  15087  efgval  15337  psrplusg  16433  psrvscafval  16442  opsrle  16524  ply1plusgfvi  16624  txindislem  17653  txhaus  17667  0met  18384  zrdivrng  22008  mbfmcst  24597  0rrv  24697  dfpo2  25367  isbnd3  26430
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877
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