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Theorem compne 31154
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 3792 . 2  |-  _V  =/=  (/)
2 ssun1 3667 . . . . . . . 8  |-  _V  C_  ( _V  u.  A
)
3 ssv 3524 . . . . . . . 8  |-  ( _V  u.  A )  C_  _V
42, 3eqssi 3520 . . . . . . 7  |-  _V  =  ( _V  u.  A
)
5 undif1 3902 . . . . . . 7  |-  ( ( _V  \  A )  u.  A )  =  ( _V  u.  A
)
64, 5eqtr4i 2499 . . . . . 6  |-  _V  =  ( ( _V  \  A )  u.  A
)
7 uneq1 3651 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  u.  A )  =  ( A  u.  A ) )
86, 7syl5eq 2520 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  _V  =  ( A  u.  A ) )
9 unidm 3647 . . . . 5  |-  ( A  u.  A )  =  A
108, 9syl6eq 2524 . . . 4  |-  ( ( _V  \  A )  =  A  ->  _V  =  A )
11 difabs 3762 . . . . . . 7  |-  ( ( _V  \  A ) 
\  A )  =  ( _V  \  A
)
12 id 22 . . . . . . 7  |-  ( ( _V  \  A )  =  A  ->  ( _V  \  A )  =  A )
1311, 12syl5req 2521 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  A  =  ( ( _V 
\  A )  \  A ) )
14 difeq1 3615 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  \  A )  =  ( A  \  A ) )
1513, 14eqtrd 2508 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  A  =  ( A  \  A ) )
16 difid 3895 . . . . 5  |-  ( A 
\  A )  =  (/)
1715, 16syl6eq 2524 . . . 4  |-  ( ( _V  \  A )  =  A  ->  A  =  (/) )
1810, 17eqtrd 2508 . . 3  |-  ( ( _V  \  A )  =  A  ->  _V  =  (/) )
1918necon3i 2707 . 2  |-  ( _V  =/=  (/)  ->  ( _V  \  A )  =/=  A
)
201, 19ax-mp 5 1  |-  ( _V 
\  A )  =/= 
A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    =/= wne 2662   _Vcvv 3113    \ cdif 3473    u. cun 3474   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786
This theorem is referenced by: (None)
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