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Theorem cnvsn0 4899
Description: The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
cnvsn0  |-  `' { (/)
}  =  (/)

Proof of Theorem cnvsn0
StepHypRef Expression
1 dfdm4 4628 . . 3  |-  dom  { (/)
}  =  ran  `' { (/) }
2 dmsn0 4898 . . 3  |-  dom  { (/)
}  =  (/)
31, 2eqtr3i 2110 . 2  |-  ran  `' { (/) }  =  (/)
4 relcnv 4810 . . 3  |-  Rel  `' { (/) }
5 relrn0 4695 . . 3  |-  ( Rel  `' { (/) }  ->  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) ) )
64, 5ax-mp 7 . 2  |-  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) )
73, 6mpbir 144 1  |-  `' { (/)
}  =  (/)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289   (/)c0 3286   {csn 3446   `'ccnv 4437   dom cdm 4438   ran crn 4439   Rel wrel 4443
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by:  brtpos0  6017  tpostpos  6029
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