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 4288	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 4925	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 324	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4282	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  ∅c0 4261  ∪ ciun 4921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-v 3433  df-dif 3886  df-nul 4262  df-iun 4923

This theorem is referenced by:  iinvdif  5009  iununi  5028  iunopeqop  5462  iunfi  9243  pwsdompw  10116  fsum2d  15724  fsumiun  15775  fprod2d  15937  prmreclem4  16881  prmreclem5  16882  fiuncmp  23387  ovolfiniun  25486  ovoliunnul  25492  finiunmbl  25529  volfiniun  25532  volsup  25541  gsumpart  33144  esum2dlem  34276  sigapildsyslem  34345  fiunelros  34358  mrsubvrs  35750  0totbnd  38140  totbndbnd  38156  fiiuncl  45513  sge0iunmptlemfi  46856  caragenfiiuncl  46958  carageniuncllem1  46964