Theorem iun0 4993

Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iun0	⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

Proof of Theorem iun0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 4268	. . . . 5 ⊢ ¬ 𝑦 ∈ ∅
2	1	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅)
3	2	nrex 3069	. . 3 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅
4		eliun 4927	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅)
5	3, 4	mtbir 325	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅
6	5	nel0 4284	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  ∅c0 4263  ∪ ciun 4923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-v 3435  df-dif 3887  df-nul 4264  df-iun 4925

This theorem is referenced by:  iunxdif3  5026  iununi  5030  funiunfv  7195  om0r  8468  kmlem11  10078  ituniiun  10340  dfrtrclrec2  15015  voliunlem1  25538  ofpreima2  32760  ssdifidllem  33541  esum2dlem  34286  sigaclfu2  34315  measvunilem0  34407  measvuni  34408  cvmscld  35514  ovolval4lem1  47104