Theorem iun0 4386

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

Proof of Theorem iun0

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3788	. . . . . 6
2	1	a1i 11	. . . . 5
3	2	nrex 2912	. . . 4
4		eliun 4335	. . . 4
5	3, 4	mtbir 299	. . 3
6	5, 1	2false 350	. 2
7	6	eqriv 2453	1

Syntax hints:  -.wn 3  =wceq 1395  e.wcel 1818  E.wrex 2808  c0 3784  U_ciun 4330

This theorem is referenced by:  iununi  4415  funiunfv  6160  om0r  7208  kmlem11  8561  ituniiun  8823  voliunlem1  21960  ofpreima2  27508  sigaclfu2  28121  measvunilem0  28184  measvuni  28185  cvmscld  28718  trpred0  29319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-nul 3785  df-iun 4332