Theorem uni0c 3762

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 3761	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3087	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 3544	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2441	. 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 1331   e. wcel 1480   A.wral 2416   C_ wss 3071   (/)c0 3363   {csn 3527   U.cuni 3736

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-uni 3737

This theorem is referenced by: (None)