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

Step	Hyp	Ref	Expression
1		vn0 3755	. 2 |- _V =/= (/)
2		ssun1 3630	. . . . . . . 8 |- _V C_ ( _V u. A )
3		ssv 3487	. . . . . . . 8 |- ( _V u. A ) C_ _V
4	2, 3	eqssi 3483	. . . . . . 7 |- _V = ( _V u. A )
5		undif1 3865	. . . . . . 7 |- ( ( _V \ A ) u. A ) = ( _V u. A )
6	4, 5	eqtr4i 2486	. . . . . 6 |- _V = ( ( _V \ A ) u. A )
7		uneq1 3614	. . . . . 6 |- ( ( _V \ A ) = A -> ( ( _V \ A ) u. A ) = ( A u. A ) )
8	6, 7	syl5eq 2507	. . . . 5 |- ( ( _V \ A ) = A -> _V = ( A u. A ) )
9		unidm 3610	. . . . 5 |- ( A u. A ) = A
10	8, 9	syl6eq 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 )
13	11, 12	syl5req 2508	. . . . . 6 |- ( ( _V \ A ) = A -> A = ( ( _V \ A ) \ A ) )
14		difeq1 3578	. . . . . 6 |- ( ( _V \ A ) = A -> ( ( _V \ A ) \ A ) = ( A \ A ) )
15	13, 14	eqtrd 2495	. . . . 5 |- ( ( _V \ A ) = A -> A = ( A \ A ) )
16		difid 3858	. . . . 5 |- ( A \ A ) = (/)
17	15, 16	syl6eq 2511	. . . 4 |- ( ( _V \ A ) = A -> A = (/) )
18	10, 17	eqtrd 2495	. . 3 |- ( ( _V \ A ) = A -> _V = (/) )
19	18	necon3i 2692	. 2 |- ( _V =/= (/) -> ( _V \ A ) =/= A )
20	1, 19	ax-mp 5	1 |- ( _V \ A ) =/= A

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)