Theorem uni0b 2519

Description: The union of a set is empty iff the set is included in the singleton of the empty set.

Assertion

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

Proof of Theorem uni0b

Step	Hyp	Ref	Expression
1		elsn 2418	. . 3 |- (x e. {(/)} <-> x = (/))
2	1	ralbii 1665	. 2 |- (A.x e. A x e. {(/)} <-> A.x e. A x = (/))
3		dfss3 2056	. 2 |- (A (_ {(/)} <-> A.x e. A x e. {(/)})
4		n0 2286	. . . 4 |- (-. U.A = (/) <-> E.y y e. U.A)
5		rexcom4 1821	. . . . 5 |- (E.x e. A E.y y e. x <-> E.yE.x e. A y e. x)
6		n0 2286	. . . . . 6 |- (-. x = (/) <-> E.y y e. x)
7	6	rexbii 1666	. . . . 5 |- (E.x e. A -. x = (/) <-> E.x e. A E.y y e. x)
8		eluni2 2503	. . . . . 6 |- (y e. U.A <-> E.x e. A y e. x)
9	8	exbii 1050	. . . . 5 |- (E.y y e. U.A <-> E.yE.x e. A y e. x)
10	5, 7, 9	3bitr4r 184	. . . 4 |- (E.y y e. U.A <-> E.x e. A -. x = (/))
11		rexnal 1652	. . . 4 |- (E.x e. A -. x = (/) <-> -. A.x e. A x = (/))
12	4, 10, 11	3bitr 177	. . 3 |- (-. U.A = (/) <-> -. A.x e. A x = (/))
13	12	con4bii 522	. 2 |- (U.A = (/) <-> A.x e. A x = (/))
14	2, 3, 13	3bitr4r 184	1 |- (U.A = (/) <-> A (_ {(/)})

Syntax hints:  -. wn 2   <-> wb 146   = wceq 955   e. wcel 957  E.wex 979  A.wral 1643  E.wrex 1644   (_ wss 2044  (/)c0 2277  {csn 2406  U.cuni 2499

This theorem is referenced by:  uni0c 2520  uni0 2521  infxpidmlem8 7519  0top 7595  cctop 7612

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-in 2048  df-ss 2050  df-nul 2278  df-sn 2409  df-uni 2500

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