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Theorem 0iin 4327
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 4272 . 2  |-  |^|_ x  e.  (/)  A  =  {
y  |  A. x  e.  (/)  y  e.  A }
2 vex 3034 . . . 4  |-  y  e. 
_V
3 ral0 3865 . . . 4  |-  A. x  e.  (/)  y  e.  A
42, 32th 247 . . 3  |-  ( y  e.  _V  <->  A. x  e.  (/)  y  e.  A
)
54abbi2i 2586 . 2  |-  _V  =  { y  |  A. x  e.  (/)  y  e.  A }
61, 5eqtr4i 2496 1  |-  |^|_ x  e.  (/)  A  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   _Vcvv 3031   (/)c0 3722   |^|_ciin 4270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-dif 3393  df-nul 3723  df-iin 4272
This theorem is referenced by:  iinrab2  4332  iinvdif  4341  riin0  4343  iin0  4575  xpriindi  4976  cmpfi  20500  ptbasfi  20673  pol0N  33545
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