MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmsn0el Structured version   Unicode version

Theorem dmsn0el 5339
Description: The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
Assertion
Ref Expression
dmsn0el  |-  ( (/)  e.  A  ->  dom  { A }  =  (/) )

Proof of Theorem dmsn0el
StepHypRef Expression
1 dmsnn0 5335 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
2 0nelelxp 4907 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  -.  (/)  e.  A
)
31, 2sylbir 205 . 2  |-  ( dom 
{ A }  =/=  (/) 
->  -.  (/)  e.  A )
43necon4ai 2663 1  |-  ( (/)  e.  A  ->  dom  { A }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   (/)c0 3628   {csn 3814    X. cxp 4876   dom cdm 4878
This theorem is referenced by:  dmsnsnsn  5348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-dm 4888
  Copyright terms: Public domain W3C validator