MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvsn0 Structured version   Visualization version   GIF version

Theorem cnvsn0 5820
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 5520 . . 3 dom {∅} = ran {∅}
2 dmsn0 5819 . . 3 dom {∅} = ∅
31, 2eqtr3i 2824 . 2 ran {∅} = ∅
4 relcnv 5721 . . 3 Rel {∅}
5 relrn0 5588 . . 3 (Rel {∅} → ({∅} = ∅ ↔ ran {∅} = ∅))
64, 5ax-mp 5 . 2 ({∅} = ∅ ↔ ran {∅} = ∅)
73, 6mpbir 223 1 {∅} = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  c0 4116  {csn 4369  ccnv 5312  dom cdm 5313  ran crn 5314  Rel wrel 5318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-br 4845  df-opab 4907  df-xp 5319  df-rel 5320  df-cnv 5321  df-dm 5323  df-rn 5324
This theorem is referenced by:  opswap  5842  brtpos0  7598  tpostpos  7611
  Copyright terms: Public domain W3C validator