Theorem compne 31590

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 3791	. 2 |- _V =/= (/)
2		ssun1 3653	. . . . . . . 8 |- _V C_ ( _V u. A )
3		ssv 3509	. . . . . . . 8 |- ( _V u. A ) C_ _V
4	2, 3	eqssi 3505	. . . . . . 7 |- _V = ( _V u. A )
5		undif1 3891	. . . . . . 7 |- ( ( _V \ A ) u. A ) = ( _V u. A )
6	4, 5	eqtr4i 2486	. . . . . 6 |- _V = ( ( _V \ A ) u. A )
7		uneq1 3637	. . . . . 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 3633	. . . . 5 |- ( A u. A ) = A
10	8, 9	syl6eq 2511	. . . 4 |- ( ( _V \ A ) = A -> _V = A )
11		difabs 3759	. . . . . . 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 3601	. . . . . 6 |- ( ( _V \ A ) = A -> ( ( _V \ A ) \ A ) = ( A \ A ) )
15	13, 14	eqtrd 2495	. . . . 5 |- ( ( _V \ A ) = A -> A = ( A \ A ) )
16		difid 3884	. . . . 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 2694	. 2 |- ( _V =/= (/) -> ( _V \ A ) =/= A )
20	1, 19	ax-mp 5	1 |- ( _V \ A ) =/= A

Syntax hints:   = wceq 1398   =/= wne 2649   _Vcvv 3106   \ cdif 3458   u. cun 3459   (/)c0 3783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784

This theorem is referenced by: (None)