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Theorem int0 4130
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 3629 . . . . . 6  |-  -.  y  e.  (/)
21pm2.21i 131 . . . . 5  |-  ( y  e.  (/)  ->  x  e.  y )
32ax-gen 1594 . . . 4  |-  A. y
( y  e.  (/)  ->  x  e.  y )
4 equid 1728 . . . 4  |-  x  =  x
53, 42th 239 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2545 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 4117 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2964 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2463 1  |-  |^| (/)  =  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1360    = wceq 1362    e. wcel 1755   {cab 2419   _Vcvv 2962   (/)c0 3625   |^|cint 4116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-v 2964  df-dif 3319  df-nul 3626  df-int 4117
This theorem is referenced by:  unissint  4140  uniintsn  4153  rint0  4156  intex  4436  intnex  4437  oev2  6951  fiint  7576  elfi2  7652  fi0  7658  cardmin2  8156  00lsp  16983  cmpfi  18852  ptbasfi  18995  fbssint  19252  fclscmp  19444  rankeq1o  28055  heibor1lem  28549
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