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Theorem int0 3843
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 3418 . . . . . 6  |-  -.  y  e.  (/)
21pm2.21i 641 . . . . 5  |-  ( y  e.  (/)  ->  x  e.  y )
32ax-gen 1442 . . . 4  |-  A. y
( y  e.  (/)  ->  x  e.  y )
4 equid 1694 . . . 4  |-  x  =  x
53, 42th 173 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2286 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 3830 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2732 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2201 1  |-  |^| (/)  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346    = wceq 1348    e. wcel 2141   {cab 2156   _Vcvv 2730   (/)c0 3414   |^|cint 3829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415  df-int 3830
This theorem is referenced by:  rint0  3868  intexr  4134  fiintim  6903  elfi2  6946  fi0  6949  bj-intexr  13905
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