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Theorem compne 27745
Description: The complement of  A is not equal to  A. (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
compne  |-  ( _V 
\  A )  =/= 
A

Proof of Theorem compne
StepHypRef Expression
1 vn0 3475 . 2  |-  _V  =/=  (/)
2 ssun1 3351 . . . . . . . 8  |-  _V  C_  ( _V  u.  A
)
3 ssv 3211 . . . . . . . 8  |-  ( _V  u.  A )  C_  _V
42, 3eqssi 3208 . . . . . . 7  |-  _V  =  ( _V  u.  A
)
5 undif1 3542 . . . . . . 7  |-  ( ( _V  \  A )  u.  A )  =  ( _V  u.  A
)
64, 5eqtr4i 2319 . . . . . 6  |-  _V  =  ( ( _V  \  A )  u.  A
)
7 uneq1 3335 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  u.  A )  =  ( A  u.  A ) )
86, 7syl5eq 2340 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  _V  =  ( A  u.  A ) )
9 unidm 3331 . . . . 5  |-  ( A  u.  A )  =  A
108, 9syl6eq 2344 . . . 4  |-  ( ( _V  \  A )  =  A  ->  _V  =  A )
11 difabs 3445 . . . . . . 7  |-  ( ( _V  \  A ) 
\  A )  =  ( _V  \  A
)
12 id 19 . . . . . . 7  |-  ( ( _V  \  A )  =  A  ->  ( _V  \  A )  =  A )
1311, 12syl5req 2341 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  A  =  ( ( _V 
\  A )  \  A ) )
14 difeq1 3300 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  \  A )  =  ( A  \  A ) )
1513, 14eqtrd 2328 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  A  =  ( A  \  A ) )
16 difid 3535 . . . . 5  |-  ( A 
\  A )  =  (/)
1715, 16syl6eq 2344 . . . 4  |-  ( ( _V  \  A )  =  A  ->  A  =  (/) )
1810, 17eqtrd 2328 . . 3  |-  ( ( _V  \  A )  =  A  ->  _V  =  (/) )
1918necon3i 2498 . 2  |-  ( _V  =/=  (/)  ->  ( _V  \  A )  =/=  A
)
201, 19ax-mp 8 1  |-  ( _V 
\  A )  =/= 
A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    =/= wne 2459   _Vcvv 2801    \ cdif 3162    u. cun 3163   (/)c0 3468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469
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