Theorem uni0c 3924

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 3923	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3217	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 3690	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2539	. 2 |- ( A. x e. A x e. { (/) } <-> A. x e. A x = (/) )
5	1, 2, 4	3bitri 206	1 |- ( U. A = (/) <-> A. x e. A x = (/) )

Syntax hints:   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   (/)c0 3496   {csn 3673   U.cuni 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-uni 3899

This theorem is referenced by: (None)