MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unissint Unicode version

Theorem unissint 3923
Description: If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 3936). (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 443 . . . . 5  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  U. A  C_  |^| A
)
2 df-ne 2481 . . . . . . 7  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 intssuni 3921 . . . . . . 7  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
42, 3sylbir 204 . . . . . 6  |-  ( -.  A  =  (/)  ->  |^| A  C_ 
U. A )
54adantl 452 . . . . 5  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  |^| A  C_  U. A
)
61, 5eqssd 3230 . . . 4  |-  ( ( U. A  C_  |^| A  /\  -.  A  =  (/) )  ->  U. A  =  |^| A )
76ex 423 . . 3  |-  ( U. A  C_  |^| A  ->  ( -.  A  =  (/)  ->  U. A  =  |^| A ) )
87orrd 367 . 2  |-  ( U. A  C_  |^| A  ->  ( A  =  (/)  \/  U. A  =  |^| A ) )
9 ssv 3232 . . . . 5  |-  U. A  C_ 
_V
10 int0 3913 . . . . 5  |-  |^| (/)  =  _V
119, 10sseqtr4i 3245 . . . 4  |-  U. A  C_ 
|^| (/)
12 inteq 3902 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
1311, 12syl5sseqr 3261 . . 3  |-  ( A  =  (/)  ->  U. A  C_ 
|^| A )
14 eqimss 3264 . . 3  |-  ( U. A  =  |^| A  ->  U. A  C_  |^| A
)
1513, 14jaoi 368 . 2  |-  ( ( A  =  (/)  \/  U. A  =  |^| A )  ->  U. A  C_  |^| A
)
168, 15impbii 180 1  |-  ( U. A  C_  |^| A  <->  ( A  =  (/)  \/  U. A  =  |^| A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1633    =/= wne 2479   _Vcvv 2822    C_ wss 3186   (/)c0 3489   U.cuni 3864   |^|cint 3899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-v 2824  df-dif 3189  df-in 3193  df-ss 3200  df-nul 3490  df-uni 3865  df-int 3900
  Copyright terms: Public domain W3C validator