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Theorem 0iun 3975
Description: An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
0iun  |-  U_ x  e.  (/)  A  =  (/)

Proof of Theorem 0iun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rex0 3481 . . . 4  |-  -.  E. x  e.  (/)  y  e.  A
2 eliun 3925 . . . 4  |-  ( y  e.  U_ x  e.  (/)  A  <->  E. x  e.  (/)  y  e.  A )
31, 2mtbir 290 . . 3  |-  -.  y  e.  U_ x  e.  (/)  A
4 noel 3472 . . 3  |-  -.  y  e.  (/)
53, 42false 339 . 2  |-  ( y  e.  U_ x  e.  (/)  A  <->  y  e.  (/) )
65eqriv 2293 1  |-  U_ x  e.  (/)  A  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   E.wrex 2557   (/)c0 3468   U_ciun 3921
This theorem is referenced by:  iununi  4002  iunfi  7160  pwsdompw  7846  fsum2d  12250  fsumiun  12295  prmreclem4  12982  prmreclem5  12983  fiuncmp  17147  ovolfiniun  18876  ovoliunnul  18882  finiunmbl  18917  volfiniun  18920  volsup  18929  0totbnd  26600  totbndbnd  26616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-nul 3469  df-iun 3923
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