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Theorem iun0 4147
Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iun0  |-  U_ x  e.  A  (/)  =  (/)

Proof of Theorem iun0
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 noel 3632 . . . . . 6  |-  -.  y  e.  (/)
21a1i 11 . . . . 5  |-  ( x  e.  A  ->  -.  y  e.  (/) )
32nrex 2808 . . . 4  |-  -.  E. x  e.  A  y  e.  (/)
4 eliun 4097 . . . 4  |-  ( y  e.  U_ x  e.  A  (/)  <->  E. x  e.  A  y  e.  (/) )
53, 4mtbir 291 . . 3  |-  -.  y  e.  U_ x  e.  A  (/)
65, 12false 340 . 2  |-  ( y  e.  U_ x  e.  A  (/)  <->  y  e.  (/) )
76eqriv 2433 1  |-  U_ x  e.  A  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   E.wrex 2706   (/)c0 3628   U_ciun 4093
This theorem is referenced by:  iununi  4175  funiunfv  5995  om0r  6783  kmlem11  8040  ituniiun  8302  voliunlem1  19444  sigaclfu2  24504  measvunilem0  24567  measvuni  24568  sibfof  24654  cvmscld  24960  trpred0  25514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-nul 3629  df-iun 4095
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