MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0iin Unicode version

Theorem 0iin 3960
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 3908 . 2  |-  |^|_ x  e.  (/)  A  =  {
y  |  A. x  e.  (/)  y  e.  A }
2 vex 2791 . . . 4  |-  y  e. 
_V
3 ral0 3558 . . . 4  |-  A. x  e.  (/)  y  e.  A
42, 32th 230 . . 3  |-  ( y  e.  _V  <->  A. x  e.  (/)  y  e.  A
)
54abbi2i 2394 . 2  |-  _V  =  { y  |  A. x  e.  (/)  y  e.  A }
61, 5eqtr4i 2306 1  |-  |^|_ x  e.  (/)  A  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   _Vcvv 2788   (/)c0 3455   |^|_ciin 3906
This theorem is referenced by:  iinrab2  3965  riin0  3975  iin0  4184  xpriindi  4822  cmpfi  17135  ptbasfi  17276  rnintintrn  25126  pol0N  30098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-dif 3155  df-nul 3456  df-iin 3908
  Copyright terms: Public domain W3C validator