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Theorem 0iun 3782
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 3298 . . . 4 ¬ ∃𝑥 ∈ ∅ 𝑦𝐴
2 eliun 3729 . . . 4 (𝑦 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦𝐴)
31, 2mtbir 631 . . 3 ¬ 𝑦 𝑥 ∈ ∅ 𝐴
4 noel 3288 . . 3 ¬ 𝑦 ∈ ∅
53, 42false 652 . 2 (𝑦 𝑥 ∈ ∅ 𝐴𝑦 ∈ ∅)
65eqriv 2085 1 𝑥 ∈ ∅ 𝐴 = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  wrex 2360  c0 3284   ciun 3725
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-nul 3285  df-iun 3727
This theorem is referenced by:  iununir  3807  rdg0  6134  iunfidisj  6634  fsum2d  10792  fsumiun  10833
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