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 3063	. . 3 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅
4		eliun 4927	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅)
5	3, 4	mtbir 323	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅
6	5	nel0 4284	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  ∃wrex 3059  ∅c0 4263  ∪ ciun 4923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-v 3429  df-dif 3888  df-nul 4264  df-iun 4925

This theorem is referenced by:  iunxdif3  5026  iununi  5030  funiunfv  7192  om0r  8463  kmlem11  10072  ituniiun  10333  dfrtrclrec2  15009  voliunlem1  25505  ofpreima2  32727  ssdifidllem  33504  esum2dlem  34224  sigaclfu2  34253  measvunilem0  34345  measvuni  34346  cvmscld  35443  ovolval4lem1  47065