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Theorem iun0 3974
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 3472 . . . . . 6  |-  -.  y  e.  (/)
21a1i 10 . . . . 5  |-  ( x  e.  A  ->  -.  y  e.  (/) )
32nrex 2658 . . . 4  |-  -.  E. x  e.  A  y  e.  (/)
4 eliun 3925 . . . 4  |-  ( y  e.  U_ x  e.  A  (/)  <->  E. x  e.  A  y  e.  (/) )
53, 4mtbir 290 . . 3  |-  -.  y  e.  U_ x  e.  A  (/)
65, 12false 339 . 2  |-  ( y  e.  U_ x  e.  A  (/)  <->  y  e.  (/) )
76eqriv 2293 1  |-  U_ x  e.  A  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   E.wrex 2557   (/)c0 3468   U_ciun 3921
This theorem is referenced by:  iununi  4002  funiunfv  5790  om0r  6554  kmlem11  7802  ituniiun  8064  voliunlem1  18923  sigaclfu2  23497  measvunilem0  23556  measvuni  23557  cvmscld  23819  trpred0  24310
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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