Theorem 0iun 4992

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 4291	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 4928	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 323	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4284	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  ∅c0 4256  ∪ ciun 4924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-nul 4257  df-iun 4926

This theorem is referenced by:  iinvdif  5009  iununi  5028  iunfi  9107  pwsdompw  9960  fsum2d  15483  fsumiun  15533  fprod2d  15691  prmreclem4  16620  prmreclem5  16621  fiuncmp  22555  ovolfiniun  24665  ovoliunnul  24671  finiunmbl  24708  volfiniun  24711  volsup  24720  gsumpart  31315  esum2dlem  32060  sigapildsyslem  32129  fiunelros  32142  mrsubvrs  33484  0totbnd  35931  totbndbnd  35947  fiiuncl  42613  sge0iunmptlemfi  43951  caragenfiiuncl  44053  carageniuncllem1  44059