Theorem 0iun 5067

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 4365	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 4999	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 323	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4359	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1536   ∈ wcel 2105  ∃wrex 3067  ∅c0 4338  ∪ ciun 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-dif 3965  df-nul 4339  df-iun 4997

This theorem is referenced by:  iinvdif  5084  iununi  5103  iunfi  9380  pwsdompw  10240  fsum2d  15803  fsumiun  15853  fprod2d  16013  prmreclem4  16952  prmreclem5  16953  fiuncmp  23427  ovolfiniun  25549  ovoliunnul  25555  finiunmbl  25592  volfiniun  25595  volsup  25604  gsumpart  33042  esum2dlem  34072  sigapildsyslem  34141  fiunelros  34154  mrsubvrs  35506  0totbnd  37759  totbndbnd  37775  fiiuncl  45004  sge0iunmptlemfi  46368  caragenfiiuncl  46470  carageniuncllem1  46476