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Theorem int0 3873
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 3441 . . . . . 6  |-  -.  y  e.  (/)
21pm2.21i 647 . . . . 5  |-  ( y  e.  (/)  ->  x  e.  y )
32ax-gen 1460 . . . 4  |-  A. y
( y  e.  (/)  ->  x  e.  y )
4 equid 1712 . . . 4  |-  x  =  x
53, 42th 174 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2305 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 3860 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2754 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2220 1  |-  |^| (/)  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1362    = wceq 1364    e. wcel 2160   {cab 2175   _Vcvv 2752   (/)c0 3437   |^|cint 3859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438  df-int 3860
This theorem is referenced by:  rint0  3898  intexr  4168  fiintim  6957  elfi2  7001  fi0  7004  bj-intexr  15118
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