MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  int0 Unicode version

Theorem int0 3876
Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
int0  |-  |^| (/)  =  _V

Proof of Theorem int0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3459 . . . . . 6  |-  -.  y  e.  (/)
21pm2.21i 123 . . . . 5  |-  ( y  e.  (/)  ->  x  e.  y )
32ax-gen 1533 . . . 4  |-  A. y
( y  e.  (/)  ->  x  e.  y )
4 eqid 2283 . . . 4  |-  x  =  x
53, 42th 230 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2395 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 3863 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2790 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2313 1  |-  |^| (/)  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   (/)c0 3455   |^|cint 3862
This theorem is referenced by:  unissint  3886  uniintsn  3899  rint0  3902  intex  4167  intnex  4168  oev2  6522  fiint  7133  elfi2  7168  fi0  7173  cardmin2  7631  incexclem  12295  incexc  12296  00lsp  15738  cmpfi  17135  ptbasfi  17276  fbssint  17533  fclscmp  17725  rankeq1o  24801  heibor1lem  26533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-nul 3456  df-int 3863
  Copyright terms: Public domain W3C validator