Theorem iun0 5025

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 4290	. . . . 5 ⊢ ¬ 𝑦 ∈ ∅
2	1	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅)
3	2	nrex 3092	. . 3 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅
4		eliun 4959	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅)
5	3, 4	mtbir 326	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅
6	5	nel0 4308	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  ∅c0 4285  ∪ ciun 4955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-v 3456  df-dif 3907  df-nul 4286  df-iun 4957

This theorem is used by:  iunxdif3  5060  iununi  5064  funiunfv  7246  om0r  8522  kmlem11  10151  ituniiun  10412  dfrtrclrec2  15102  ssdifidllem  21495  voliunlem1  25720  ofpreima2  33022  esum2dlem  34491  sigaclfu2  34520  measvunilem0  34612  measvuni  34613  cvmscld  35773  ovolval4lem1  47391