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Theorem cnvsn0 6045
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 5741 . . 3 dom {∅} = ran {∅}
2 dmsn0 6044 . . 3 dom {∅} = ∅
31, 2eqtr3i 2847 . 2 ran {∅} = ∅
4 relcnv 5945 . . 3 Rel {∅}
5 relrn0 5818 . . 3 (Rel {∅} → ({∅} = ∅ ↔ ran {∅} = ∅))
64, 5ax-mp 5 . 2 ({∅} = ∅ ↔ ran {∅} = ∅)
73, 6mpbir 234 1 {∅} = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  c0 4265  {csn 4539  ccnv 5531  dom cdm 5532  ran crn 5533  Rel wrel 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-dm 5542  df-rn 5543
This theorem is referenced by:  opswap  6064  brtpos0  7886  tpostpos  7899
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