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Theorem 0iin 4331
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 4276 . 2  |-  |^|_ x  e.  (/)  A  =  {
y  |  A. x  e.  (/)  y  e.  A }
2 vex 3064 . . . 4  |-  y  e. 
_V
3 ral0 3880 . . . 4  |-  A. x  e.  (/)  y  e.  A
42, 32th 241 . . 3  |-  ( y  e.  _V  <->  A. x  e.  (/)  y  e.  A
)
54abbi2i 2537 . 2  |-  _V  =  { y  |  A. x  e.  (/)  y  e.  A }
61, 5eqtr4i 2436 1  |-  |^|_ x  e.  (/)  A  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407    e. wcel 1844   {cab 2389   A.wral 2756   _Vcvv 3061   (/)c0 3740   |^|_ciin 4274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-v 3063  df-dif 3419  df-nul 3741  df-iin 4276
This theorem is referenced by:  iinrab2  4336  iinvdif  4345  riin0  4347  iin0  4570  xpriindi  4962  cmpfi  20203  ptbasfi  20376  pol0N  32939
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