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Theorem compne 29864
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 3755 . 2  |-  _V  =/=  (/)
2 ssun1 3630 . . . . . . . 8  |-  _V  C_  ( _V  u.  A
)
3 ssv 3487 . . . . . . . 8  |-  ( _V  u.  A )  C_  _V
42, 3eqssi 3483 . . . . . . 7  |-  _V  =  ( _V  u.  A
)
5 undif1 3865 . . . . . . 7  |-  ( ( _V  \  A )  u.  A )  =  ( _V  u.  A
)
64, 5eqtr4i 2486 . . . . . 6  |-  _V  =  ( ( _V  \  A )  u.  A
)
7 uneq1 3614 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  u.  A )  =  ( A  u.  A ) )
86, 7syl5eq 2507 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  _V  =  ( A  u.  A ) )
9 unidm 3610 . . . . 5  |-  ( A  u.  A )  =  A
108, 9syl6eq 2511 . . . 4  |-  ( ( _V  \  A )  =  A  ->  _V  =  A )
11 difabs 3725 . . . . . . 7  |-  ( ( _V  \  A ) 
\  A )  =  ( _V  \  A
)
12 id 22 . . . . . . 7  |-  ( ( _V  \  A )  =  A  ->  ( _V  \  A )  =  A )
1311, 12syl5req 2508 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  A  =  ( ( _V 
\  A )  \  A ) )
14 difeq1 3578 . . . . . 6  |-  ( ( _V  \  A )  =  A  ->  (
( _V  \  A
)  \  A )  =  ( A  \  A ) )
1513, 14eqtrd 2495 . . . . 5  |-  ( ( _V  \  A )  =  A  ->  A  =  ( A  \  A ) )
16 difid 3858 . . . . 5  |-  ( A 
\  A )  =  (/)
1715, 16syl6eq 2511 . . . 4  |-  ( ( _V  \  A )  =  A  ->  A  =  (/) )
1810, 17eqtrd 2495 . . 3  |-  ( ( _V  \  A )  =  A  ->  _V  =  (/) )
1918necon3i 2692 . 2  |-  ( _V  =/=  (/)  ->  ( _V  \  A )  =/=  A
)
201, 19ax-mp 5 1  |-  ( _V 
\  A )  =/= 
A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    =/= wne 2648   _Vcvv 3078    \ cdif 3436    u. cun 3437   (/)c0 3748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749
This theorem is referenced by: (None)
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