MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vn0 Unicode version

Theorem vn0 3595
Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
Assertion
Ref Expression
vn0  |-  _V  =/=  (/)

Proof of Theorem vn0
StepHypRef Expression
1 vex 2919 . 2  |-  x  e. 
_V
2 ne0i 3594 . 2  |-  ( x  e.  _V  ->  _V  =/=  (/) )
31, 2ax-mp 8 1  |-  _V  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1721    =/= wne 2567   _Vcvv 2916   (/)c0 3588
This theorem is referenced by:  uniintsn  4047  relrelss  5352  imasaddfnlem  13708  imasvscafn  13717  cmpfi  17425  fclscmp  18015  compne  27510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-nul 3589
  Copyright terms: Public domain W3C validator