Theorem 0iun 5063

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 4995	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 323	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4354	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  ∅c0 4333  ∪ ciun 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-nul 4334  df-iun 4993

This theorem is referenced by:  iinvdif  5080  iununi  5099  iunfi  9383  pwsdompw  10243  fsum2d  15807  fsumiun  15857  fprod2d  16017  prmreclem4  16957  prmreclem5  16958  fiuncmp  23412  ovolfiniun  25536  ovoliunnul  25542  finiunmbl  25579  volfiniun  25582  volsup  25591  gsumpart  33060  esum2dlem  34093  sigapildsyslem  34162  fiunelros  34175  mrsubvrs  35527  0totbnd  37780  totbndbnd  37796  fiiuncl  45070  sge0iunmptlemfi  46428  caragenfiiuncl  46530  carageniuncllem1  46536