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Theorem unissint 4074
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4087). (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 444 . . . . 5  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  U. A  C_  |^| A
)
2 df-ne 2601 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 intssuni 4072 . . . . . . 7  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
42, 3sylbir 205 . . . . . 6  |-  ( -.  A  =  (/)  ->  |^| A  C_ 
U. A )
54adantl 453 . . . . 5  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  |^| A  C_  U. A
)
61, 5eqssd 3365 . . . 4  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  U. A  =  |^| A )
76ex 424 . . 3  |-  ( U. A  C_  |^| A  ->  ( -.  A  =  (/)  ->  U. A  =  |^| A ) )
87orrd 368 . 2  |-  ( U. A  C_  |^| A  ->  ( A  =  (/)  \/  U. A  =  |^| A ) )
9 ssv 3368 . . . . 5  |-  U. A  C_ 
_V
10 int0 4064 . . . . 5  |-  |^| (/)  =  _V
119, 10sseqtr4i 3381 . . . 4  |-  U. A  C_ 
|^| (/)
12 inteq 4053 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
1311, 12syl5sseqr 3397 . . 3  |-  ( A  =  (/)  ->  U. A  C_ 
|^| A )
14 eqimss 3400 . . 3  |-  ( U. A  =  |^| A  ->  U. A  C_  |^| A
)
1513, 14jaoi 369 . 2  |-  ( ( A  =  (/)  \/  U. A  =  |^| A )  ->  U. A  C_  |^| A
)
168, 15impbii 181 1  |-  ( U. A  C_  |^| A  <->  ( A  =  (/)  \/  U. A  =  |^| A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    =/= wne 2599   _Vcvv 2956    C_ wss 3320   (/)c0 3628   U.cuni 4015   |^|cint 4050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-uni 4016  df-int 4051
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