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Theorem cnvsn0 6209
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 5893 . . 3 dom {∅} = ran {∅}
2 dmsn0 6208 . . 3 dom {∅} = ∅
31, 2eqtr3i 2758 . 2 ran {∅} = ∅
4 relcnv 6103 . . 3 Rel {∅}
5 relrn0 5967 . . 3 (Rel {∅} → ({∅} = ∅ ↔ ran {∅} = ∅))
64, 5ax-mp 5 . 2 ({∅} = ∅ ↔ ran {∅} = ∅)
73, 6mpbir 230 1 {∅} = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  c0 4319  {csn 4625  ccnv 5672  dom cdm 5673  ran crn 5674  Rel wrel 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5144  df-opab 5206  df-xp 5679  df-rel 5680  df-cnv 5681  df-dm 5683  df-rn 5684
This theorem is referenced by:  opswap  6228  brtpos0  8233  tpostpos  8246
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