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Theorem 0iun 4961
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 𝑥 ∈ ∅ 𝐴 = ∅

Proof of Theorem 0iun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rex0 4289 . . 3 ¬ ∃𝑥 ∈ ∅ 𝑦𝐴
2 eliun 4898 . . 3 (𝑦 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦𝐴)
31, 2mtbir 326 . 2 ¬ 𝑦 𝑥 ∈ ∅ 𝐴
43nel0 4283 1 𝑥 ∈ ∅ 𝐴 = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2114  wrex 3131  c0 4265   ciun 4894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-nul 4266  df-iun 4896
This theorem is referenced by:  iinvdif  4977  iununi  4996  iunfi  8800  pwsdompw  9615  fsum2d  15117  fsumiun  15167  fprod2d  15326  prmreclem4  16244  prmreclem5  16245  fiuncmp  22007  ovolfiniun  24103  ovoliunnul  24109  finiunmbl  24146  volfiniun  24149  volsup  24158  esum2dlem  31425  sigapildsyslem  31494  fiunelros  31507  mrsubvrs  32843  0totbnd  35169  totbndbnd  35185  fiiuncl  41633  sge0iunmptlemfi  42991  caragenfiiuncl  43093  carageniuncllem1  43099
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