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Theorem int0 3708
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 3291 . . . . . 6  |-  -.  y  e.  (/)
21pm2.21i 611 . . . . 5  |-  ( y  e.  (/)  ->  x  e.  y )
32ax-gen 1384 . . . 4  |-  A. y
( y  e.  (/)  ->  x  e.  y )
4 equid 1635 . . . 4  |-  x  =  x
53, 42th 173 . . 3  |-  ( A. y ( y  e.  (/)  ->  x  e.  y )  <->  x  =  x
)
65abbii 2204 . 2  |-  { x  |  A. y ( y  e.  (/)  ->  x  e.  y ) }  =  { x  |  x  =  x }
7 df-int 3695 . 2  |-  |^| (/)  =  {
x  |  A. y
( y  e.  (/)  ->  x  e.  y ) }
8 df-v 2622 . 2  |-  _V  =  { x  |  x  =  x }
96, 7, 83eqtr4i 2119 1  |-  |^| (/)  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1288    = wceq 1290    e. wcel 1439   {cab 2075   _Vcvv 2620   (/)c0 3287   |^|cint 3694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-nul 3288  df-int 3695
This theorem is referenced by:  rint0  3733  intexr  3992  fiintim  6693  bj-intexr  12072
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