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Theorem unissint 4299
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4312). (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 457 . . . . 5  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  U. A  C_  |^| A
)
2 df-ne 2657 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 intssuni 4297 . . . . . . 7  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
42, 3sylbir 213 . . . . . 6  |-  ( -.  A  =  (/)  ->  |^| A  C_ 
U. A )
54adantl 466 . . . . 5  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  |^| A  C_  U. A
)
61, 5eqssd 3514 . . . 4  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  U. A  =  |^| A )
76ex 434 . . 3  |-  ( U. A  C_  |^| A  ->  ( -.  A  =  (/)  ->  U. A  =  |^| A ) )
87orrd 378 . 2  |-  ( U. A  C_  |^| A  ->  ( A  =  (/)  \/  U. A  =  |^| A ) )
9 ssv 3517 . . . . 5  |-  U. A  C_ 
_V
10 int0 4289 . . . . 5  |-  |^| (/)  =  _V
119, 10sseqtr4i 3530 . . . 4  |-  U. A  C_ 
|^| (/)
12 inteq 4278 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
1311, 12syl5sseqr 3546 . . 3  |-  ( A  =  (/)  ->  U. A  C_ 
|^| A )
14 eqimss 3549 . . 3  |-  ( U. A  =  |^| A  ->  U. A  C_  |^| A
)
1513, 14jaoi 379 . 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 368    /\ wa 369    = wceq 1374    =/= wne 2655   _Vcvv 3106    C_ wss 3469   (/)c0 3778   U.cuni 4238   |^|cint 4275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-v 3108  df-dif 3472  df-in 3476  df-ss 3483  df-nul 3779  df-uni 4239  df-int 4276
This theorem is referenced by: (None)
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