Theorem uni0c 3822

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 3821	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3137	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 3600	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2476	. 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 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   (/)c0 3414   {csn 3583   U.cuni 3796

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-uni 3797

This theorem is referenced by: (None)