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Theorem 0iin 4383
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 4328 . 2  |-  |^|_ x  e.  (/)  A  =  {
y  |  A. x  e.  (/)  y  e.  A }
2 vex 3116 . . . 4  |-  y  e. 
_V
3 ral0 3932 . . . 4  |-  A. x  e.  (/)  y  e.  A
42, 32th 239 . . 3  |-  ( y  e.  _V  <->  A. x  e.  (/)  y  e.  A
)
54abbi2i 2600 . 2  |-  _V  =  { y  |  A. x  e.  (/)  y  e.  A }
61, 5eqtr4i 2499 1  |-  |^|_ x  e.  (/)  A  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   _Vcvv 3113   (/)c0 3785   |^|_ciin 4326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-dif 3479  df-nul 3786  df-iin 4328
This theorem is referenced by:  iinrab2  4388  iinvdif  4397  riin0  4399  iin0  4621  xpriindi  5137  cmpfi  19674  ptbasfi  19817  pol0N  34705
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