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 4283	. . . . 5 ⊢ ¬ 𝑦 ∈ ∅
2	1	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅)
3	2	nrex 3090	. . 3 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅
4		eliun 4954	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅)
5	3, 4	mtbir 326	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅
6	5	nel0 4301	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  ∅c0 4278  ∪ ciun 4950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-dif 4047  df-nul 4279  df-iun 4952

This theorem is used by:  iunxdif3  5054  iununi  5058  funiunfv  7252  om0r  8547  kmlem11  10239  ituniiun  10500  dfrtrclrec2  15211  ssdifidllem  21640  voliunlem1  25871  ofpreima2  33260  esum2dlem  34724  sigaclfu2  34753  measvunilem0  34846  measvuni  34847  cvmscld  36038  ovolval4lem1  47659