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Theorem 0iin 4354
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 4299 . 2  |-  |^|_ x  e.  (/)  A  =  {
y  |  A. x  e.  (/)  y  e.  A }
2 vex 3084 . . . 4  |-  y  e. 
_V
3 ral0 3902 . . . 4  |-  A. x  e.  (/)  y  e.  A
42, 32th 242 . . 3  |-  ( y  e.  _V  <->  A. x  e.  (/)  y  e.  A
)
54abbi2i 2555 . 2  |-  _V  =  { y  |  A. x  e.  (/)  y  e.  A }
61, 5eqtr4i 2454 1  |-  |^|_ x  e.  (/)  A  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1868   {cab 2407   A.wral 2775   _Vcvv 3081   (/)c0 3761   |^|_ciin 4297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-v 3083  df-dif 3439  df-nul 3762  df-iin 4299
This theorem is referenced by:  iinrab2  4359  iinvdif  4368  riin0  4370  iin0  4594  xpriindi  4986  cmpfi  20407  ptbasfi  20580  pol0N  33390
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