Theorem iun0 4576

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	|- U_ x e. A (/) = (/)

Proof of Theorem iun0

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3919	. . . . 5 |- -. y e. (/)
2	1	a1i 11	. . . 4 |- ( x e. A -> -. y e. (/) )
3	2	nrex 3000	. . 3 |- -. E. x e. A y e. (/)
4		eliun 4524	. . 3 |- ( y e. U_ x e. A (/) <-> E. x e. A y e. (/) )
5	3, 4	mtbir 313	. 2 |- -. y e. U_ x e. A (/)
6	5	nel0 3932	1 |- U_ x e. A (/) = (/)

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   E.wrex 2913   (/)c0 3915   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916  df-iun 4522

This theorem is referenced by:  iunxdif3  4606  iununi  4610  funiunfv  6506  om0r  7619  kmlem11  8982  ituniiun  9244  voliunlem1  23318  ofpreima2  29466  esum2dlem  30154  sigaclfu2  30184  measvunilem0  30276  measvuni  30277  cvmscld  31255  trpred0  31736  ovolval4lem1  40863