MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmsn0 Structured version   Unicode version

Theorem dmsn0 5458
Description: The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
Assertion
Ref Expression
dmsn0  |-  dom  { (/)
}  =  (/)

Proof of Theorem dmsn0
StepHypRef Expression
1 0nelxp 5016 . 2  |-  -.  (/)  e.  ( _V  X.  _V )
2 dmsnn0 5456 . . 3  |-  ( (/)  e.  ( _V  X.  _V ) 
<->  dom  { (/) }  =/=  (/) )
32necon2bbii 2721 . 2  |-  ( dom 
{ (/) }  =  (/)  <->  -.  (/) 
e.  ( _V  X.  _V ) )
41, 3mpbir 209 1  |-  dom  { (/)
}  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   {csn 4016    X. cxp 4986   dom cdm 4988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-dm 4998
This theorem is referenced by:  cnvsn0  5459  dmsnopss  5463  1st0  6779  2nd0  6780
  Copyright terms: Public domain W3C validator