Theorem uni0c 3679

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 3678	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3015	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 3463	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2384	. 2 |- ( A. x e. A x e. { (/) } <-> A. x e. A x = (/) )
5	1, 2, 4	3bitri 204	1 |- ( U. A = (/) <-> A. x e. A x = (/) )

Syntax hints:   <-> wb 103   = wceq 1289   e. wcel 1438   A.wral 2359   C_ wss 2999   (/)c0 3286   {csn 3446   U.cuni 3653

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-uni 3654

This theorem is referenced by: (None)