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Theorem cnvsn0 5014
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 4738 . . 3  |-  dom  { (/)
}  =  ran  `' { (/) }
2 dmsn0 5013 . . 3  |-  dom  { (/)
}  =  (/)
31, 2eqtr3i 2163 . 2  |-  ran  `' { (/) }  =  (/)
4 relcnv 4924 . . 3  |-  Rel  `' { (/) }
5 relrn0 4808 . . 3  |-  ( Rel  `' { (/) }  ->  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) ) )
64, 5ax-mp 5 . 2  |-  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) )
73, 6mpbir 145 1  |-  `' { (/)
}  =  (/)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1332   (/)c0 3367   {csn 3531   `'ccnv 4545   dom cdm 4546   ran crn 4547   Rel wrel 4551
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553  df-cnv 4554  df-dm 4556  df-rn 4557
This theorem is referenced by:  brtpos0  6156  tpostpos  6168
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