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Theorem int0 3774
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 3366 . . . . . 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 2253 . . . 4  |-  x  =  x
53, 42th 232 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2361 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 3761 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2729 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2283 1  |-  |^| (/)  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532    = wceq 1619    e. wcel 1621   {cab 2239   _Vcvv 2727   (/)c0 3362   |^|cint 3760
This theorem is referenced by:  unissint  3784  uniintsn  3797  rint0  3800  intex  4065  intnex  4066  oev2  6408  fiint  7018  elfi2  7052  fi0  7057  cardmin2  7515  00lsp  15573  cmpfi  16967  ptbasfi  17108  fbssint  17365  fclscmp  17557  rankeq1o  23975  heibor1lem  25699
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-dif 3081  df-nul 3363  df-int 3761
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