Theorem uni0c 3837

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 3836	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3147	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 3611	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2483	. 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 1353   e. wcel 2148   A.wral 2455   C_ wss 3131   (/)c0 3424   {csn 3594   U.cuni 3811

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-uni 3812

This theorem is referenced by: (None)