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Theorem xp0 5097
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 4767 . . 3  |-  ( (/)  X.  A )  =  (/)
21cnveqi 4855 . 2  |-  `' (
(/)  X.  A )  =  `' (/)
3 cnvxp 5096 . 2  |-  `' (
(/)  X.  A )  =  ( A  X.  (/) )
4 cnv0 5083 . 2  |-  `' (/)  =  (/)
52, 3, 43eqtr3i 2312 1  |-  ( A  X.  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1624   (/)c0 3456    X. cxp 4686   `'ccnv 4687
This theorem is referenced by:  xpnz  5098  xpdisj2  5101  dmxpss  5106  rnxpid  5108  xpcan  5111  unixp  5203  fconst5  5692  difxp1  6115  dfac5lem3  7747  xpcdaen  7804  fpwwe2lem13  8259  comfffval  13595  fuchom  13829  xpccofval  13950  frmdplusg  14470  mulgfval  14562  mulgfvi  14565  ga0  14746  symgplusg  14770  efgval  15020  psrplusg  16120  psrvscafval  16129  opsrle  16211  ply1plusgfvi  16314  txindislem  17321  txhaus  17335  0met  17924  zrdivrng  21091  dfpo2  23515  fixpc  24492  isbnd3  25907
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696
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