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Theorem cnvsn0 5467
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 5186 . . 3  |-  dom  { (/)
}  =  ran  `' { (/) }
2 dmsn0 5466 . . 3  |-  dom  { (/)
}  =  (/)
31, 2eqtr3i 2491 . 2  |-  ran  `' { (/) }  =  (/)
4 relcnv 5365 . . 3  |-  Rel  `' { (/) }
5 relrn0 5251 . . 3  |-  ( Rel  `' { (/) }  ->  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) ) )
64, 5ax-mp 5 . 2  |-  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) )
73, 6mpbir 209 1  |-  `' { (/)
}  =  (/)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1374   (/)c0 3778   {csn 4020   `'ccnv 4991   dom cdm 4992   ran crn 4993   Rel wrel 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003
This theorem is referenced by:  opswap  5486  brtpos0  6952  tpostpos  6965
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