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Theorem 0iun 3985
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 3478 . . . 4 ¬ ∃𝑥 ∈ ∅ 𝑦𝐴
2 eliun 3931 . . . 4 (𝑦 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦𝐴)
31, 2mtbir 673 . . 3 ¬ 𝑦 𝑥 ∈ ∅ 𝐴
4 noel 3464 . . 3 ¬ 𝑦 ∈ ∅
53, 42false 703 . 2 (𝑦 𝑥 ∈ ∅ 𝐴𝑦 ∈ ∅)
65eqriv 2202 1 𝑥 ∈ ∅ 𝐴 = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1373  wcel 2176  wrex 2485  c0 3460   ciun 3927
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-nul 3461  df-iun 3929
This theorem is referenced by:  iununir  4011  rdg0  6473  iunfidisj  7048  fsum2d  11746  fsumiun  11788  fprod2d  11934  iuncld  14587
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