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Theorem 0iin 3976
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 3924 . 2  |-  |^|_ x  e.  (/)  A  =  {
y  |  A. x  e.  (/)  y  e.  A }
2 vex 2804 . . . 4  |-  y  e. 
_V
3 ral0 3571 . . . 4  |-  A. x  e.  (/)  y  e.  A
42, 32th 230 . . 3  |-  ( y  e.  _V  <->  A. x  e.  (/)  y  e.  A
)
54abbi2i 2407 . 2  |-  _V  =  { y  |  A. x  e.  (/)  y  e.  A }
61, 5eqtr4i 2319 1  |-  |^|_ x  e.  (/)  A  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   _Vcvv 2801   (/)c0 3468   |^|_ciin 3922
This theorem is referenced by:  iinrab2  3981  riin0  3991  iin0  4200  xpriindi  4838  cmpfi  17151  ptbasfi  17292  rnintintrn  25229  pol0N  30720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-dif 3168  df-nul 3469  df-iin 3924
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