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Theorem 0iin 4216
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 4162 . 2  |-  |^|_ x  e.  (/)  A  =  {
y  |  A. x  e.  (/)  y  e.  A }
2 vex 2965 . . . 4  |-  y  e. 
_V
3 ral0 3772 . . . 4  |-  A. x  e.  (/)  y  e.  A
42, 32th 239 . . 3  |-  ( y  e.  _V  <->  A. x  e.  (/)  y  e.  A
)
54abbi2i 2544 . 2  |-  _V  =  { y  |  A. x  e.  (/)  y  e.  A }
61, 5eqtr4i 2456 1  |-  |^|_ x  e.  (/)  A  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362    e. wcel 1755   {cab 2419   A.wral 2705   _Vcvv 2962   (/)c0 3625   |^|_ciin 4160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-v 2964  df-dif 3319  df-nul 3626  df-iin 4162
This theorem is referenced by:  iinrab2  4221  iinvdif  4230  riin0  4232  iin0  4454  xpriindi  4963  cmpfi  18853  ptbasfi  18996  pol0N  33126
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