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Theorem cnvsn0 5141
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 4872 . . 3  |-  dom  { (/)
}  =  ran  `' { (/) }
2 dmsn0 5140 . . 3  |-  dom  { (/)
}  =  (/)
31, 2eqtr3i 2305 . 2  |-  ran  `' { (/) }  =  (/)
4 relcnv 5051 . . 3  |-  Rel  `' { (/) }
5 relrn0 4937 . . 3  |-  ( Rel  `' { (/) }  ->  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) ) )
64, 5ax-mp 8 . 2  |-  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) )
73, 6mpbir 200 1  |-  `' { (/)
}  =  (/)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   (/)c0 3455   {csn 3640   `'ccnv 4688   dom cdm 4689   ran crn 4690   Rel wrel 4694
This theorem is referenced by:  opswap  5159  brtpos0  6241  tpostpos  6254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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