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Theorem 0iun 3874
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 3381 . . . 4  |-  -.  E. x  e.  (/)  y  e.  A
2 eliun 3821 . . . 4  |-  ( y  e.  U_ x  e.  (/)  A  <->  E. x  e.  (/)  y  e.  A )
31, 2mtbir 661 . . 3  |-  -.  y  e.  U_ x  e.  (/)  A
4 noel 3368 . . 3  |-  -.  y  e.  (/)
53, 42false 691 . 2  |-  ( y  e.  U_ x  e.  (/)  A  <->  y  e.  (/) )
65eqriv 2137 1  |-  U_ x  e.  (/)  A  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1332    e. wcel 1481   E.wrex 2418   (/)c0 3364   U_ciun 3817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-dif 3074  df-nul 3365  df-iun 3819
This theorem is referenced by:  iununir  3900  rdg0  6288  iunfidisj  6838  fsum2d  11232  fsumiun  11274  iuncld  12314
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