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Theorem xp0 4773
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 4446 . . 3  |-  ( (/)  X.  A )  =  (/)
21cnveqi 4538 . 2  |-  `' (
(/)  X.  A )  =  `' (/)
3 cnvxp 4772 . 2  |-  `' (
(/)  X.  A )  =  ( A  X.  (/) )
4 cnv0 4757 . 2  |-  `' (/)  =  (/)
52, 3, 43eqtr3i 2110 1  |-  ( A  X.  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1285   (/)c0 3258    X. cxp 4369   `'ccnv 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379
This theorem is referenced by:  xpeq0r  4776  xpdisj2  4778
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