Theorem uni0c 3861

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 3860	. 2 |- ( U. A = (/) <-> A C_ { (/) } )
2		dfss3 3169	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
3		velsn 3635	. . 3 |- ( x e. { (/) } <-> x = (/) )
4	3	ralbii 2500	. 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 1364   e. wcel 2164   A.wral 2472   C_ wss 3153   (/)c0 3446   {csn 3618   U.cuni 3835

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-uni 3836

This theorem is referenced by: (None)