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Theorem iun0 4371
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 3774 . . . . . 6  |-  -.  y  e.  (/)
21a1i 11 . . . . 5  |-  ( x  e.  A  ->  -.  y  e.  (/) )
32nrex 2898 . . . 4  |-  -.  E. x  e.  A  y  e.  (/)
4 eliun 4320 . . . 4  |-  ( y  e.  U_ x  e.  A  (/)  <->  E. x  e.  A  y  e.  (/) )
53, 4mtbir 299 . . 3  |-  -.  y  e.  U_ x  e.  A  (/)
65, 12false 350 . 2  |-  ( y  e.  U_ x  e.  A  (/)  <->  y  e.  (/) )
76eqriv 2439 1  |-  U_ x  e.  A  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1383    e. wcel 1804   E.wrex 2794   (/)c0 3770   U_ciun 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-v 3097  df-dif 3464  df-nul 3771  df-iun 4317
This theorem is referenced by:  iununi  4400  funiunfv  6145  om0r  7191  kmlem11  8543  ituniiun  8805  voliunlem1  21833  ofpreima2  27380  sigaclfu2  27994  measvunilem0  28057  measvuni  28058  cvmscld  28591  trpred0  29294
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