Theorem uni0c 3798

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 3797	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3118	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 3577	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2463	. 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 1335   e. wcel 2128   A.wral 2435   C_ wss 3102   (/)c0 3394   {csn 3560   U.cuni 3772

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-uni 3773

This theorem is referenced by: (None)