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Theorem 0iun 3984
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 3477 . . . 4 ¬ ∃𝑥 ∈ ∅ 𝑦𝐴
2 eliun 3930 . . . 4 (𝑦 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦𝐴)
31, 2mtbir 672 . . 3 ¬ 𝑦 𝑥 ∈ ∅ 𝐴
4 noel 3463 . . 3 ¬ 𝑦 ∈ ∅
53, 42false 702 . 2 (𝑦 𝑥 ∈ ∅ 𝐴𝑦 ∈ ∅)
65eqriv 2201 1 𝑥 ∈ ∅ 𝐴 = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1372  wcel 2175  wrex 2484  c0 3459   ciun 3926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-nul 3460  df-iun 3928
This theorem is referenced by:  iununir  4010  rdg0  6472  iunfidisj  7047  fsum2d  11717  fsumiun  11759  fprod2d  11905  iuncld  14558
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