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Theorem cnvsn0 4839
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 4575 . . 3  |-  dom  { (/)
}  =  ran  `' { (/) }
2 dmsn0 4838 . . 3  |-  dom  { (/)
}  =  (/)
31, 2eqtr3i 2105 . 2  |-  ran  `' { (/) }  =  (/)
4 relcnv 4753 . . 3  |-  Rel  `' { (/) }
5 relrn0 4642 . . 3  |-  ( Rel  `' { (/) }  ->  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) ) )
64, 5ax-mp 7 . 2  |-  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) )
73, 6mpbir 144 1  |-  `' { (/)
}  =  (/)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285   (/)c0 3267   {csn 3416   `'ccnv 4390   dom cdm 4391   ran crn 4392   Rel wrel 4396
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 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by:  brtpos0  5922  tpostpos  5934
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