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Theorem 0iin 3783
Description: An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
0iin  |-  |^|_ x  e.  (/)  A  =  _V

Proof of Theorem 0iin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-iin 3728 . 2  |-  |^|_ x  e.  (/)  A  =  {
y  |  A. x  e.  (/)  y  e.  A }
2 vex 2622 . . . 4  |-  y  e. 
_V
3 ral0 3379 . . . 4  |-  A. x  e.  (/)  y  e.  A
42, 32th 172 . . 3  |-  ( y  e.  _V  <->  A. x  e.  (/)  y  e.  A
)
54abbi2i 2202 . 2  |-  _V  =  { y  |  A. x  e.  (/)  y  e.  A }
61, 5eqtr4i 2111 1  |-  |^|_ x  e.  (/)  A  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   _Vcvv 2619   (/)c0 3284   |^|_ciin 3726
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-ral 2364  df-v 2621  df-dif 2999  df-nul 3285  df-iin 3728
This theorem is referenced by:  riin0  3796  iin0r  3996
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