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Theorem unissint 4224
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4237). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
unissint  |-  ( U. A  C_  |^| A  <->  ( A  =  (/)  \/  U. A  =  |^| A ) )

Proof of Theorem unissint
StepHypRef Expression
1 simpl 455 . . . . 5  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  U. A  C_  |^| A
)
2 df-ne 2579 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 intssuni 4222 . . . . . . 7  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
42, 3sylbir 213 . . . . . 6  |-  ( -.  A  =  (/)  ->  |^| A  C_ 
U. A )
54adantl 464 . . . . 5  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  |^| A  C_  U. A
)
61, 5eqssd 3434 . . . 4  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  U. A  =  |^| A )
76ex 432 . . 3  |-  ( U. A  C_  |^| A  ->  ( -.  A  =  (/)  ->  U. A  =  |^| A ) )
87orrd 376 . 2  |-  ( U. A  C_  |^| A  ->  ( A  =  (/)  \/  U. A  =  |^| A ) )
9 ssv 3437 . . . . 5  |-  U. A  C_ 
_V
10 int0 4213 . . . . 5  |-  |^| (/)  =  _V
119, 10sseqtr4i 3450 . . . 4  |-  U. A  C_ 
|^| (/)
12 inteq 4202 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
1311, 12syl5sseqr 3466 . . 3  |-  ( A  =  (/)  ->  U. A  C_ 
|^| A )
14 eqimss 3469 . . 3  |-  ( U. A  =  |^| A  ->  U. A  C_  |^| A
)
1513, 14jaoi 377 . 2  |-  ( ( A  =  (/)  \/  U. A  =  |^| A )  ->  U. A  C_  |^| A
)
168, 15impbii 188 1  |-  ( U. A  C_  |^| A  <->  ( A  =  (/)  \/  U. A  =  |^| A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1399    =/= wne 2577   _Vcvv 3034    C_ wss 3389   (/)c0 3711   U.cuni 4163   |^|cint 4199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-v 3036  df-dif 3392  df-in 3396  df-ss 3403  df-nul 3712  df-uni 4164  df-int 4200
This theorem is referenced by: (None)
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