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Theorem cnvsn0 5178
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 4909 . . 3  |-  dom  { (/)
}  =  ran  `' { (/) }
2 dmsn0 5177 . . 3  |-  dom  { (/)
}  =  (/)
31, 2eqtr3i 2338 . 2  |-  ran  `' { (/) }  =  (/)
4 relcnv 5088 . . 3  |-  Rel  `' { (/) }
5 relrn0 4974 . . 3  |-  ( Rel  `' { (/) }  ->  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) ) )
64, 5ax-mp 8 . 2  |-  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) )
73, 6mpbir 200 1  |-  `' { (/)
}  =  (/)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1633   (/)c0 3489   {csn 3674   `'ccnv 4725   dom cdm 4726   ran crn 4727   Rel wrel 4731
This theorem is referenced by:  opswap  5196  brtpos0  6283  tpostpos  6296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-cnv 4734  df-dm 4736  df-rn 4737
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