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Theorem cnvsn0 5294
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 5018 . . 3  |-  dom  { (/)
}  =  ran  `' { (/) }
2 dmsn0 5293 . . 3  |-  dom  { (/)
}  =  (/)
31, 2eqtr3i 2435 . 2  |-  ran  `' { (/) }  =  (/)
4 relcnv 5197 . . 3  |-  Rel  `' { (/) }
5 relrn0 5083 . . 3  |-  ( Rel  `' { (/) }  ->  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) ) )
64, 5ax-mp 5 . 2  |-  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) )
73, 6mpbir 211 1  |-  `' { (/)
}  =  (/)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    = wceq 1407   (/)c0 3740   {csn 3974   `'ccnv 4824   dom cdm 4825   ran crn 4826   Rel wrel 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-rel 4832  df-cnv 4833  df-dm 4835  df-rn 4836
This theorem is referenced by:  opswap  5313  brtpos0  6967  tpostpos  6980
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