Theorem uni0c 3815

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 3814	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3132	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 3593	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2472	. 2 |- ( A. x e. A x e. { (/) } <-> A. x e. A x = (/) )
5	1, 2, 4	3bitri 205	1 |- ( U. A = (/) <-> A. x e. A x = (/) )

Syntax hints:   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   (/)c0 3409   {csn 3576   U.cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-uni 3790

This theorem is referenced by: (None)