Theorem uni0b 4463

Description: The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)

Assertion

Ref	Expression
uni0b	|- ( U. A = (/) <-> A C_ { (/) } )

Proof of Theorem uni0b

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 4193	. . 3 |- ( x e. { (/) } <-> x = (/) )
2	1	ralbii 2980	. 2 |- ( A. x e. A x e. { (/) } <-> A. x e. A x = (/) )
3		dfss3 3592	. 2 |- ( A C_ { (/) } <-> A. x e. A x e. { (/) } )
4		neq0 3930	. . . 4 |- ( -. U. A = (/) <-> E. y y e. U. A )
5		rexcom4 3225	. . . . 5 |- ( E. x e. A E. y y e. x <-> E. y E. x e. A y e. x )
6		neq0 3930	. . . . . 6 |- ( -. x = (/) <-> E. y y e. x )
7	6	rexbii 3041	. . . . 5 |- ( E. x e. A -. x = (/) <-> E. x e. A E. y y e. x )
8		eluni2 4440	. . . . . 6 |- ( y e. U. A <-> E. x e. A y e. x )
9	8	exbii 1774	. . . . 5 |- ( E. y y e. U. A <-> E. y E. x e. A y e. x )
10	5, 7, 9	3bitr4ri 293	. . . 4 |- ( E. y y e. U. A <-> E. x e. A -. x = (/) )
11		rexnal 2995	. . . 4 |- ( E. x e. A -. x = (/) <-> -. A. x e. A x = (/) )
12	4, 10, 11	3bitri 286	. . 3 |- ( -. U. A = (/) <-> -. A. x e. A x = (/) )
13	12	con4bii 311	. 2 |- ( U. A = (/) <-> A. x e. A x = (/) )
14	2, 3, 13	3bitr4ri 293	1 |- ( U. A = (/) <-> A C_ { (/) } )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437

This theorem is referenced by:  uni0c  4464  uni0  4465  fin1a2lem11  9232  zornn0g  9327  0top  20787  filconn  21687  alexsubALTlem2  21852  ordcmp  32446  unisn0  39222