Theorem 0iun 5071

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 4360	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 5005	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 322	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4353	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1534   ∈ wcel 2099  ∃wrex 3060  ∅c0 4325  ∪ ciun 5001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-v 3464  df-dif 3950  df-nul 4326  df-iun 5003

This theorem is referenced by:  iinvdif  5088  iununi  5107  iunfi  9385  pwsdompw  10247  fsum2d  15775  fsumiun  15825  fprod2d  15983  prmreclem4  16921  prmreclem5  16922  fiuncmp  23399  ovolfiniun  25521  ovoliunnul  25527  finiunmbl  25564  volfiniun  25567  volsup  25576  gsumpart  32923  esum2dlem  33925  sigapildsyslem  33994  fiunelros  34007  mrsubvrs  35350  0totbnd  37474  totbndbnd  37490  fiiuncl  44666  sge0iunmptlemfi  46034  caragenfiiuncl  46136  carageniuncllem1  46142