Theorem 0iun 5021

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 4308	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 4955	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 326	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4302	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  ∅c0 4279  ∪ ciun 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-dif 3902  df-nul 4280  df-iun 4953

This theorem is used by:  iinvdif  5040  iununi  5059  iunopeqop  5498  iunfi  9313  pwsdompw  10208  fsum2d  15860  fsumiun  15911  fprod2d  16071  prmreclem4  17014  prmreclem5  17015  fiuncmp  23632  ovolfiniun  25732  ovoliunnul  25738  finiunmbl  25775  volfiniun  25778  volsup  25787  gsumpart  33506  esum2dlem  34605  sigapildsyslem  34675  fiunelros  34688  mrsubvrs  36104  0totbnd  38526  totbndbnd  38542  fiiuncl  45902  sge0iunmptlemfi  47244  caragenfiiuncl  47346  carageniuncllem1  47352