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Theorem 0iun 3877
 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 3384 . . . 4 ¬ ∃𝑥 ∈ ∅ 𝑦𝐴
2 eliun 3824 . . . 4 (𝑦 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦𝐴)
31, 2mtbir 661 . . 3 ¬ 𝑦 𝑥 ∈ ∅ 𝐴
4 noel 3371 . . 3 ¬ 𝑦 ∈ ∅
53, 42false 691 . 2 (𝑦 𝑥 ∈ ∅ 𝐴𝑦 ∈ ∅)
65eqriv 2137 1 𝑥 ∈ ∅ 𝐴 = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∈ wcel 1481  ∃wrex 2418  ∅c0 3367  ∪ ciun 3820 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 2691  df-dif 3077  df-nul 3368  df-iun 3822 This theorem is referenced by:  iununir  3903  rdg0  6291  iunfidisj  6841  fsum2d  11235  fsumiun  11277  iuncld  12321
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