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Theorem int0 3817
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 3401 . . . . . 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 3804 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2742 . 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 2740   (/)c0 3397   |^|cint 3803
This theorem is referenced by:  unissint  3827  uniintsn  3840  rint0  3843  intex  4109  intnex  4110  oev2  6455  fiint  7066  elfi2  7101  fi0  7106  cardmin2  7564  00lsp  15665  cmpfi  17062  ptbasfi  17203  fbssint  17460  fclscmp  17652  rankeq1o  24141  heibor1lem  25865
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 2742  df-dif 3097  df-nul 3398  df-int 3804
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