Theorem 0iun 4986

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 4317	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 4923	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 325	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4311	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  ∅c0 4291  ∪ ciun 4919

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3939  df-nul 4292  df-iun 4921

This theorem is referenced by:  iinvdif  5002  iununi  5021  iunfi  8812  pwsdompw  9626  fsum2d  15126  fsumiun  15176  fprod2d  15335  prmreclem4  16255  prmreclem5  16256  fiuncmp  22012  ovolfiniun  24102  ovoliunnul  24108  finiunmbl  24145  volfiniun  24148  volsup  24157  esum2dlem  31351  sigapildsyslem  31420  fiunelros  31433  mrsubvrs  32769  0totbnd  35066  totbndbnd  35082  fiiuncl  41347  sge0iunmptlemfi  42715  caragenfiiuncl  42817  carageniuncllem1  42823