Theorem iunn0 4390

Description: There is a nonempty class in an indexed collection (x) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iunn0

Distinct variable group:   ,

Proof of Theorem iunn0

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 3129	. . 3
2		eliun 4335	. . . 4
3	2	exbii 1667	. . 3
4	1, 3	bitr4i 252	. 2
5		n0 3794	. . 3
6	5	rexbii 2959	. 2
7		n0 3794	. 2
8	4, 6, 7	3bitr4i 277	1

Syntax hints:  <->wb 184  E.wex 1612  e.wcel 1818  =/=wne 2652  E.wrex 2808  c0 3784  U_ciun 4330

This theorem is referenced by:  fsuppmapnn0fiubex  12098  lbsextlem2  17805

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-or 370  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-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-nul 3785  df-iun 4332