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Theorem xp0 4926
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 0xp 4587 . . 3  |-  ( (/)  X.  A )  =  (/)
21cnveqi 4682 . 2  |-  `' (
(/)  X.  A )  =  `' (/)
3 cnvxp 4925 . 2  |-  `' (
(/)  X.  A )  =  ( A  X.  (/) )
4 cnv0 4910 . 2  |-  `' (/)  =  (/)
52, 3, 43eqtr3i 2144 1  |-  ( A  X.  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1314   (/)c0 3331    X. cxp 4505   `'ccnv 4506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515
This theorem is referenced by:  xpeq0r  4929  xpdisj2  4932  djuassen  7037  xpdjuen  7038  0met  12448
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