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Theorem cnvsn0 6200
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 5886 . . 3 dom {∅} = ran {∅}
2 dmsn0 6199 . . 3 dom {∅} = ∅
31, 2eqtr3i 2754 . 2 ran {∅} = ∅
4 relcnv 6094 . . 3 Rel {∅}
5 relrn0 5959 . . 3 (Rel {∅} → ({∅} = ∅ ↔ ran {∅} = ∅))
64, 5ax-mp 5 . 2 ({∅} = ∅ ↔ ran {∅} = ∅)
73, 6mpbir 230 1 {∅} = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  c0 4315  {csn 4621  ccnv 5666  dom cdm 5667  ran crn 5668  Rel wrel 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678
This theorem is referenced by:  opswap  6219  brtpos0  8214  tpostpos  8227
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