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Theorem int0 3836
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
StepHypRef Expression
1 noel 3420 . . . . . 6  |-  -.  y  e.  (/)
21pm2.21i 125 . . . . 5  |-  ( y  e.  (/)  ->  x  e.  y )
32ax-gen 1536 . . . 4  |-  A. y
( y  e.  (/)  ->  x  e.  y )
4 eqid 2256 . . . 4  |-  x  =  x
53, 42th 232 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2368 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 3823 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2759 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2286 1  |-  |^| (/)  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532    = wceq 1619    e. wcel 1621   {cab 2242   _Vcvv 2757   (/)c0 3416   |^|cint 3822
This theorem is referenced by:  unissint  3846  uniintsn  3859  rint0  3862  intex  4129  intnex  4130  oev2  6476  fiint  7087  elfi2  7122  fi0  7127  cardmin2  7585  00lsp  15686  cmpfi  17083  ptbasfi  17224  fbssint  17481  fclscmp  17673  rankeq1o  24162  heibor1lem  25886
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-dif 3116  df-nul 3417  df-int 3823
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