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

Step	Hyp	Ref	Expression
1		vn0 3792	. 2 |- _V =/= (/)
2		ssun1 3667	. . . . . . . 8 |- _V C_ ( _V u. A )
3		ssv 3524	. . . . . . . 8 |- ( _V u. A ) C_ _V
4	2, 3	eqssi 3520	. . . . . . 7 |- _V = ( _V u. A )
5		undif1 3902	. . . . . . 7 |- ( ( _V \ A ) u. A ) = ( _V u. A )
6	4, 5	eqtr4i 2499	. . . . . 6 |- _V = ( ( _V \ A ) u. A )
7		uneq1 3651	. . . . . 6 |- ( ( _V \ A ) = A -> ( ( _V \ A ) u. A ) = ( A u. A ) )
8	6, 7	syl5eq 2520	. . . . 5 |- ( ( _V \ A ) = A -> _V = ( A u. A ) )
9		unidm 3647	. . . . 5 |- ( A u. A ) = A
10	8, 9	syl6eq 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 )
13	11, 12	syl5req 2521	. . . . . 6 |- ( ( _V \ A ) = A -> A = ( ( _V \ A ) \ A ) )
14		difeq1 3615	. . . . . 6 |- ( ( _V \ A ) = A -> ( ( _V \ A ) \ A ) = ( A \ A ) )
15	13, 14	eqtrd 2508	. . . . 5 |- ( ( _V \ A ) = A -> A = ( A \ A ) )
16		difid 3895	. . . . 5 |- ( A \ A ) = (/)
17	15, 16	syl6eq 2524	. . . 4 |- ( ( _V \ A ) = A -> A = (/) )
18	10, 17	eqtrd 2508	. . 3 |- ( ( _V \ A ) = A -> _V = (/) )
19	18	necon3i 2707	. 2 |- ( _V =/= (/) -> ( _V \ A ) =/= A )
20	1, 19	ax-mp 5	1 |- ( _V \ A ) =/= A

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)