Theorem iun0 4985

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 4296	. . . . 5 ⊢ ¬ 𝑦 ∈ ∅
2	1	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑦 ∈ ∅)
3	2	nrex 3269	. . 3 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅
4		eliun 4923	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅ ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∅)
5	3, 4	mtbir 325	. 2 ⊢ ¬ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∅
6	5	nel0 4311	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  ∅c0 4291  ∪ ciun 4919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3939  df-nul 4292  df-iun 4921

This theorem is referenced by:  iunxdif3  5017  iununi  5021  funiunfv  7007  om0r  8164  kmlem11  9586  ituniiun  9844  voliunlem1  24151  ofpreima2  30411  esum2dlem  31351  sigaclfu2  31380  measvunilem0  31472  measvuni  31473  cvmscld  32520  trpred0  33075  ovolval4lem1  42951