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Theorem int0 3755
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 3337 . . . . . 6  |-  -.  y  e.  (/)
21pm2.21i 620 . . . . 5  |-  ( y  e.  (/)  ->  x  e.  y )
32ax-gen 1410 . . . 4  |-  A. y
( y  e.  (/)  ->  x  e.  y )
4 equid 1662 . . . 4  |-  x  =  x
53, 42th 173 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2233 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 3742 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2662 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2148 1  |-  |^| (/)  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1314    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660   (/)c0 3333   |^|cint 3741
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-nul 3334  df-int 3742
This theorem is referenced by:  rint0  3780  intexr  4045  fiintim  6785  elfi2  6828  fi0  6831  bj-intexr  13033
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