Theorem 0iun 5009

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 4307	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 4943	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 323	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4301	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  ∅c0 4280  ∪ ciun 4939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-v 3438  df-dif 3900  df-nul 4281  df-iun 4941

This theorem is referenced by:  iinvdif  5026  iununi  5045  iunfi  9227  pwsdompw  10094  fsum2d  15678  fsumiun  15728  fprod2d  15888  prmreclem4  16831  prmreclem5  16832  fiuncmp  23319  ovolfiniun  25429  ovoliunnul  25435  finiunmbl  25472  volfiniun  25475  volsup  25484  gsumpart  33037  esum2dlem  34105  sigapildsyslem  34174  fiunelros  34187  mrsubvrs  35566  0totbnd  37823  totbndbnd  37839  fiiuncl  45172  sge0iunmptlemfi  46521  caragenfiiuncl  46623  carageniuncllem1  46629