Theorem 0iun 5065

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.

Step	Hyp	Ref	Expression
1		rex0 4356	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 5000	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 322	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4349	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  ∅c0 4321  ∪ ciun 4996

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-v 3476  df-dif 3950  df-nul 4322  df-iun 4998

This theorem is referenced by:  iinvdif  5082  iununi  5101  iunfi  9336  pwsdompw  10195  fsum2d  15713  fsumiun  15763  fprod2d  15921  prmreclem4  16848  prmreclem5  16849  fiuncmp  22899  ovolfiniun  25009  ovoliunnul  25015  finiunmbl  25052  volfiniun  25055  volsup  25064  gsumpart  32194  esum2dlem  33078  sigapildsyslem  33147  fiunelros  33160  mrsubvrs  34501  0totbnd  36629  totbndbnd  36645  fiiuncl  43737  sge0iunmptlemfi  45115  caragenfiiuncl  45217  carageniuncllem1  45223