Theorem uni0c 3875

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 3874	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3181	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 3649	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2511	. 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 1372   e. wcel 2175   A.wral 2483   C_ wss 3165   (/)c0 3459   {csn 3632   U.cuni 3849

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-uni 3850

This theorem is referenced by: (None)