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Theorem compne 27642
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 3462 . 2  |-  _V  =/=  (/)
2 ssun1 3338 . . . . . . . 8  |-  _V  C_  ( _V  u.  A
)
3 ssv 3198 . . . . . . . 8  |-  ( _V  u.  A )  C_  _V
42, 3eqssi 3195 . . . . . . 7  |-  _V  =  ( _V  u.  A
)
5 undif1 3529 . . . . . . 7  |-  ( ( _V  \  A )  u.  A )  =  ( _V  u.  A
)
64, 5eqtr4i 2306 . . . . . 6  |-  _V  =  ( ( _V  \  A )  u.  A
)
7 uneq1 3322 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  u.  A )  =  ( A  u.  A ) )
86, 7syl5eq 2327 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  _V  =  ( A  u.  A ) )
9 unidm 3318 . . . . 5  |-  ( A  u.  A )  =  A
108, 9syl6eq 2331 . . . 4  |-  ( ( _V  \  A )  =  A  ->  _V  =  A )
11 difabs 3432 . . . . . . 7  |-  ( ( _V  \  A ) 
\  A )  =  ( _V  \  A
)
12 id 19 . . . . . . 7  |-  ( ( _V  \  A )  =  A  ->  ( _V  \  A )  =  A )
1311, 12syl5req 2328 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  A  =  ( ( _V 
\  A )  \  A ) )
14 difeq1 3287 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  \  A )  =  ( A  \  A ) )
1513, 14eqtrd 2315 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  A  =  ( A  \  A ) )
16 difid 3522 . . . . 5  |-  ( A 
\  A )  =  (/)
1715, 16syl6eq 2331 . . . 4  |-  ( ( _V  \  A )  =  A  ->  A  =  (/) )
1810, 17eqtrd 2315 . . 3  |-  ( ( _V  \  A )  =  A  ->  _V  =  (/) )
1918necon3i 2485 . 2  |-  ( _V  =/=  (/)  ->  ( _V  \  A )  =/=  A
)
201, 19ax-mp 8 1  |-  ( _V 
\  A )  =/= 
A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    =/= wne 2446   _Vcvv 2788    \ cdif 3149    u. cun 3150   (/)c0 3455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456
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