Theorem unissint 3241

Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 3253). (The proof was shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
unissint	|- (U.A C_ |^|A <-> (A = (/) \/ U.A = |^|A))

Proof of Theorem unissint

Step	Hyp	Ref	Expression
1		simpl 346	. . . . 5 |- ((U.A C_ |^|A /\ -. A = (/)) -> U.A C_ |^|A)
2		df-ne 2019	. . . . . . 7 |- (A =/= (/) <-> -. A = (/))
3		intssuni 3240	. . . . . . 7 |- (A =/= (/) -> |^|A C_ U.A)
4	2, 3	sylbir 218	. . . . . 6 |- (-. A = (/) -> |^|A C_ U.A)
5	4	adantl 424	. . . . 5 |- ((U.A C_ |^|A /\ -. A = (/)) -> |^|A C_ U.A)
6	1, 5	eqssd 2633	. . . 4 |- ((U.A C_ |^|A /\ -. A = (/)) -> U.A = |^|A)
7	6	ex 402	. . 3 |- (U.A C_ |^|A -> (-. A = (/) -> U.A = |^|A))
8	7	orrd 250	. 2 |- (U.A C_ |^|A -> (A = (/) \/ U.A = |^|A))
9		ssv 2636	. . . . 5 |- U.A C_ _V
10		int0 3230	. . . . 5 |- |^|(/) = _V
11	9, 10	sseqtr4i 2650	. . . 4 |- U.A C_ |^|(/)
12		inteq 3217	. . . . 5 |- (A = (/) -> |^|A = |^|(/))
13	12	sseq2d 2645	. . . 4 |- (A = (/) -> (U.A C_ |^|A <-> U.A C_ |^|(/)))
14	11, 13	mpbiri 211	. . 3 |- (A = (/) -> U.A C_ |^|A)
15		eqimss 2665	. . 3 |- (U.A = |^|A -> U.A C_ |^|A)
16	14, 15	jaoi 368	. 2 |- ((A = (/) \/ U.A = |^|A) -> U.A C_ |^|A)
17	8, 16	impbii 174	1 |- (U.A C_ |^|A <-> (A = (/) \/ U.A = |^|A))

Syntax hints:  -. wn 2   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  U.cuni 3177  |^|cint 3214

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-uni 3178  df-int 3215

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