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Theorem iun0 4310
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 3725 . . . . . 6  |-  -.  y  e.  (/)
21a1i 11 . . . . 5  |-  ( x  e.  A  ->  -.  y  e.  (/) )
32nrex 2900 . . . 4  |-  -.  E. x  e.  A  y  e.  (/)
4 eliun 4259 . . . 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 2446 1  |-  U_ x  e.  A  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1757   E.wrex 2793   (/)c0 3721   U_ciun 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ral 2797  df-rex 2798  df-v 3056  df-dif 3415  df-nul 3722  df-iun 4257
This theorem is referenced by:  iununi  4339  funiunfv  6050  om0r  7065  kmlem11  8416  ituniiun  8678  voliunlem1  21133  ofpreima2  26105  sigaclfu2  26684  measvunilem0  26747  measvuni  26748  cvmscld  27282  trpred0  27820
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