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Theorem cnvsn0 6166
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 5855 . . 3 dom {∅} = ran {∅}
2 dmsn0 6165 . . 3 dom {∅} = ∅
31, 2eqtr3i 2763 . 2 ran {∅} = ∅
4 relcnv 6060 . . 3 Rel {∅}
5 relrn0 5928 . . 3 (Rel {∅} → ({∅} = ∅ ↔ ran {∅} = ∅))
64, 5ax-mp 5 . 2 ({∅} = ∅ ↔ ran {∅} = ∅)
73, 6mpbir 230 1 {∅} = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  c0 4286  {csn 4590  ccnv 5636  dom cdm 5637  ran crn 5638  Rel wrel 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648
This theorem is referenced by:  opswap  6185  brtpos0  8168  tpostpos  8181
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