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

Theorem cnvsn0 5572
 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 5286 . . 3 dom {∅} = ran {∅}
2 dmsn0 5571 . . 3 dom {∅} = ∅
31, 2eqtr3i 2645 . 2 ran {∅} = ∅
4 relcnv 5472 . . 3 Rel {∅}
5 relrn0 5353 . . 3 (Rel {∅} → ({∅} = ∅ ↔ ran {∅} = ∅))
64, 5ax-mp 5 . 2 ({∅} = ∅ ↔ ran {∅} = ∅)
73, 6mpbir 221 1 {∅} = ∅
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1480  ∅c0 3897  {csn 4155  ◡ccnv 5083  dom cdm 5084  ran crn 5085  Rel wrel 5089 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095 This theorem is referenced by:  opswap  5591  brtpos0  7319  tpostpos  7332
 Copyright terms: Public domain W3C validator