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Theorem unissint 4296
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4309). (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 2640 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 intssuni 4294 . . . . . . 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 3506 . . . 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 3509 . . . . 5  |-  U. A  C_ 
_V
10 int0 4285 . . . . 5  |-  |^| (/)  =  _V
119, 10sseqtr4i 3522 . . . 4  |-  U. A  C_ 
|^| (/)
12 inteq 4274 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
1311, 12syl5sseqr 3538 . . 3  |-  ( A  =  (/)  ->  U. A  C_ 
|^| A )
14 eqimss 3541 . . 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 1383    =/= wne 2638   _Vcvv 3095    C_ wss 3461   (/)c0 3770   U.cuni 4234   |^|cint 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-v 3097  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3771  df-uni 4235  df-int 4272
This theorem is referenced by: (None)
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