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

Theorem 0iin 4151
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 4098 . 2  |-  |^|_ x  e.  (/)  A  =  {
y  |  A. x  e.  (/)  y  e.  A }
2 vex 2961 . . . 4  |-  y  e. 
_V
3 ral0 3734 . . . 4  |-  A. x  e.  (/)  y  e.  A
42, 32th 232 . . 3  |-  ( y  e.  _V  <->  A. x  e.  (/)  y  e.  A
)
54abbi2i 2549 . 2  |-  _V  =  { y  |  A. x  e.  (/)  y  e.  A }
61, 5eqtr4i 2461 1  |-  |^|_ x  e.  (/)  A  =  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   _Vcvv 2958   (/)c0 3630   |^|_ciin 4096
This theorem is referenced by:  iinrab2  4156  riin0  4166  iin0  4375  xpriindi  5013  cmpfi  17473  ptbasfi  17615  pol0N  30768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-dif 3325  df-nul 3631  df-iin 4098
  Copyright terms: Public domain W3C validator