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Theorem compne 27621
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 3637 . 2  |-  _V  =/=  (/)
2 ssun1 3512 . . . . . . . 8  |-  _V  C_  ( _V  u.  A
)
3 ssv 3370 . . . . . . . 8  |-  ( _V  u.  A )  C_  _V
42, 3eqssi 3366 . . . . . . 7  |-  _V  =  ( _V  u.  A
)
5 undif1 3705 . . . . . . 7  |-  ( ( _V  \  A )  u.  A )  =  ( _V  u.  A
)
64, 5eqtr4i 2461 . . . . . 6  |-  _V  =  ( ( _V  \  A )  u.  A
)
7 uneq1 3496 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  u.  A )  =  ( A  u.  A ) )
86, 7syl5eq 2482 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  _V  =  ( A  u.  A ) )
9 unidm 3492 . . . . 5  |-  ( A  u.  A )  =  A
108, 9syl6eq 2486 . . . 4  |-  ( ( _V  \  A )  =  A  ->  _V  =  A )
11 difabs 3607 . . . . . . 7  |-  ( ( _V  \  A ) 
\  A )  =  ( _V  \  A
)
12 id 21 . . . . . . 7  |-  ( ( _V  \  A )  =  A  ->  ( _V  \  A )  =  A )
1311, 12syl5req 2483 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  A  =  ( ( _V 
\  A )  \  A ) )
14 difeq1 3460 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  \  A )  =  ( A  \  A ) )
1513, 14eqtrd 2470 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  A  =  ( A  \  A ) )
16 difid 3698 . . . . 5  |-  ( A 
\  A )  =  (/)
1715, 16syl6eq 2486 . . . 4  |-  ( ( _V  \  A )  =  A  ->  A  =  (/) )
1810, 17eqtrd 2470 . . 3  |-  ( ( _V  \  A )  =  A  ->  _V  =  (/) )
1918necon3i 2645 . 2  |-  ( _V  =/=  (/)  ->  ( _V  \  A )  =/=  A
)
201, 19ax-mp 8 1  |-  ( _V 
\  A )  =/= 
A
Colors of variables: wff set class
Syntax hints:    = wceq 1653    =/= wne 2601   _Vcvv 2958    \ cdif 3319    u. cun 3320   (/)c0 3630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631
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