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Theorem int0 3697
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 3288 . . . . . 6  |-  -.  y  e.  (/)
21pm2.21i 610 . . . . 5  |-  ( y  e.  (/)  ->  x  e.  y )
32ax-gen 1383 . . . 4  |-  A. y
( y  e.  (/)  ->  x  e.  y )
4 equid 1634 . . . 4  |-  x  =  x
53, 42th 172 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2203 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 3684 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2621 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2118 1  |-  |^| (/)  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287    = wceq 1289    e. wcel 1438   {cab 2074   _Vcvv 2619   (/)c0 3284   |^|cint 3683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-nul 3285  df-int 3684
This theorem is referenced by:  rint0  3722  intexr  3978  bj-intexr  11456
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