Theorem 0iun 4711

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 4085	. . 3 ⊢ ¬ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴
2		eliun 4658	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴 ↔ ∃𝑥 ∈ ∅ 𝑦 ∈ 𝐴)
3	1, 2	mtbir 312	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ ∅ 𝐴
4	3	nel0 4079	1 ⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅

Syntax hints:   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  ∅c0 4063  ∪ ciun 4654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-nul 4064  df-iun 4656

This theorem is referenced by:  iinvdif  4726  iununi  4744  iunfi  8410  pwsdompw  9228  fsum2d  14710  fsumiun  14760  fprod2d  14918  prmreclem4  15830  prmreclem5  15831  fiuncmp  21428  ovolfiniun  23489  ovoliunnul  23495  finiunmbl  23532  volfiniun  23535  volsup  23544  esum2dlem  30494  sigapildsyslem  30564  fiunelros  30577  mrsubvrs  31757  0totbnd  33904  totbndbnd  33920  fiiuncl  39755  sge0iunmptlemfi  41147  caragenfiiuncl  41249  carageniuncllem1  41255