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Theorem int0 3785
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 3367 . . . . . 6  |-  -.  y  e.  (/)
21pm2.21i 635 . . . . 5  |-  ( y  e.  (/)  ->  x  e.  y )
32ax-gen 1425 . . . 4  |-  A. y
( y  e.  (/)  ->  x  e.  y )
4 equid 1677 . . . 4  |-  x  =  x
53, 42th 173 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2255 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 3772 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2688 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2170 1  |-  |^| (/)  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1329    = wceq 1331    e. wcel 1480   {cab 2125   _Vcvv 2686   (/)c0 3363   |^|cint 3771
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364  df-int 3772
This theorem is referenced by:  rint0  3810  intexr  4075  fiintim  6817  elfi2  6860  fi0  6863  bj-intexr  13159
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