Theorem unissintOLD 3242

Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 3253).

Assertion

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

Proof of Theorem unissintOLD

Step	Hyp	Ref	Expression
1		sspss 2707	. 2 |- (U.A C_ |^|A <-> (U.A C. |^|A \/ U.A = |^|A))
2		dfpss3 2695	. . . . 5 |- (U.A C. |^|A <-> (U.A C_ |^|A /\ -. |^|A C_ U.A))
3		intssuni 3240	. . . . . . 7 |- (A =/= (/) -> |^|A C_ U.A)
4	3	necon1bi 2048	. . . . . 6 |- (-. |^|A C_ U.A -> A = (/))
5	4	adantl 424	. . . . 5 |- ((U.A C_ |^|A /\ -. |^|A C_ U.A) -> A = (/))
6	2, 5	sylbi 216	. . . 4 |- (U.A C. |^|A -> A = (/))
7		vn0 2882	. . . . . . . 8 |- _V =/= (/)
8		necom 2094	. . . . . . . 8 |- (_V =/= (/) <-> (/) =/= _V)
9	7, 8	mpbi 206	. . . . . . 7 |- (/) =/= _V
10		df-ne 2019	. . . . . . 7 |- ((/) =/= _V <-> -. (/) = _V)
11	9, 10	mpbi 206	. . . . . 6 |- -. (/) = _V
12		pssv 2913	. . . . . 6 |- ((/) C. _V <-> -. (/) = _V)
13	11, 12	mpbir 207	. . . . 5 |- (/) C. _V
14		unieq 3185	. . . . . . 7 |- (A = (/) -> U.A = U.(/))
15		uni0 3205	. . . . . . 7 |- U.(/) = (/)
16	14, 15	syl6eq 1944	. . . . . 6 |- (A = (/) -> U.A = (/))
17		inteq 3217	. . . . . . 7 |- (A = (/) -> |^|A = |^|(/))
18		int0 3230	. . . . . . 7 |- |^|(/) = _V
19	17, 18	syl6eq 1944	. . . . . 6 |- (A = (/) -> |^|A = _V)
20	16, 19	psseq12d 2704	. . . . 5 |- (A = (/) -> (U.A C. |^|A <-> (/) C. _V))
21	13, 20	mpbiri 211	. . . 4 |- (A = (/) -> U.A C. |^|A)
22	6, 21	impbii 174	. . 3 |- (U.A C. |^|A <-> A = (/))
23	22	orbi1i 276	. 2 |- ((U.A C. |^|A \/ U.A = |^|A) <-> (A = (/) \/ U.A = |^|A))
24	1, 23	bitri 190	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   C. wpss 2594  (/)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-pss 2607  df-nul 2876  df-sn 3049  df-uni 3178  df-int 3215

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