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Theorem 0iun 3770
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 3289 . . . 4  |-  -.  E. x  e.  (/)  y  e.  A
2 eliun 3717 . . . 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 3279 . . 3  |-  -.  y  e.  (/)
53, 42false 650 . 2  |-  ( y  e.  U_ x  e.  (/)  A  <->  y  e.  (/) )
65eqriv 2082 1  |-  U_ x  e.  (/)  A  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1287    e. wcel 1436   E.wrex 2356   (/)c0 3275   U_ciun 3713
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-nul 3276  df-iun 3715
This theorem is referenced by:  iununir  3794  rdg0  6106
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