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Theorem 0iun 3755
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 3281 . . . 4  |-  -.  E. x  e.  (/)  y  e.  A
2 eliun 3702 . . . 4  |-  ( y  e.  U_ x  e.  (/)  A  <->  E. x  e.  (/)  y  e.  A )
31, 2mtbir 629 . . 3  |-  -.  y  e.  U_ x  e.  (/)  A
4 noel 3271 . . 3  |-  -.  y  e.  (/)
53, 42false 650 . 2  |-  ( y  e.  U_ x  e.  (/)  A  <->  y  e.  (/) )
65eqriv 2080 1  |-  U_ x  e.  (/)  A  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   E.wrex 2354   (/)c0 3267   U_ciun 3698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-nul 3268  df-iun 3700
This theorem is referenced by:  iununir  3779  rdg0  6056
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