Theorem iun0 5019

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 4292	. . . . 5 ⊢ ¬ 𝑦 ∈ ∅
2	1	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅)
3	2	nrex 3066	. . 3 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅
4		eliun 4952	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅)
5	3, 4	mtbir 323	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅
6	5	nel0 4308	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ∅c0 4287  ∪ ciun 4948

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-v 3444  df-dif 3906  df-nul 4288  df-iun 4950

This theorem is referenced by:  iunxdif3  5052  iununi  5056  funiunfv  7206  om0r  8478  kmlem11  10085  ituniiun  10346  dfrtrclrec2  14995  voliunlem1  25524  ofpreima2  32762  ssdifidllem  33555  esum2dlem  34276  sigaclfu2  34305  measvunilem0  34397  measvuni  34398  cvmscld  35495  ovolval4lem1  47036