Theorem uni0c 4464

Description: The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)

Assertion

Ref	Expression
uni0c	|- ( U. A = (/) <-> A. x e. A x = (/) )

Distinct variable group:   x, A

Proof of Theorem uni0c

Step	Hyp	Ref	Expression
1		uni0b 4463	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3592	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 4193	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2980	. 2 |- ( A. x e. A x e. { (/) } <-> A. x e. A x = (/) )
5	1, 2, 4	3bitri 286	1 |- ( U. A = (/) <-> A. x e. A x = (/) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436

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-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437

This theorem is referenced by:  fin1a2lem13  9234  fctop  20808  cctop  20810