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Theorem cnvsn0 5416
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 5141 . . 3  |-  dom  { (/)
}  =  ran  `' { (/) }
2 dmsn0 5415 . . 3  |-  dom  { (/)
}  =  (/)
31, 2eqtr3i 2485 . 2  |-  ran  `' { (/) }  =  (/)
4 relcnv 5315 . . 3  |-  Rel  `' { (/) }
5 relrn0 5206 . . 3  |-  ( Rel  `' { (/) }  ->  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) ) )
64, 5ax-mp 5 . 2  |-  ( `' { (/) }  =  (/)  <->  ran  `' { (/) }  =  (/) )
73, 6mpbir 209 1  |-  `' { (/)
}  =  (/)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370   (/)c0 3746   {csn 3986   `'ccnv 4948   dom cdm 4949   ran crn 4950   Rel wrel 4954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-cnv 4957  df-dm 4959  df-rn 4960
This theorem is referenced by:  opswap  5435  brtpos0  6863  tpostpos  6876
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