Theorem compne 38643

Description: The complement of A is not equal to A. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.)

Assertion

Ref	Expression
compne	|- ( _V \ A ) =/= A

Proof of Theorem compne

Step	Hyp	Ref	Expression
1		vn0 3924	. 2 |- _V =/= (/)
2		id 22	. . . . . . 7 |- ( ( _V \ A ) = A -> ( _V \ A ) = A )
3		difeq1 3721	. . . . . . . 8 |- ( ( _V \ A ) = A -> ( ( _V \ A ) \ A ) = ( A \ A ) )
4		difabs 3892	. . . . . . . 8 |- ( ( _V \ A ) \ A ) = ( _V \ A )
5		difid 3948	. . . . . . . 8 |- ( A \ A ) = (/)
6	3, 4, 5	3eqtr3g 2679	. . . . . . 7 |- ( ( _V \ A ) = A -> ( _V \ A ) = (/) )
7	2, 6	eqtr3d 2658	. . . . . 6 |- ( ( _V \ A ) = A -> A = (/) )
8	7	difeq2d 3728	. . . . 5 |- ( ( _V \ A ) = A -> ( _V \ A ) = ( _V \ (/) ) )
9		dif0 3950	. . . . 5 |- ( _V \ (/) ) = _V
10	8, 9	syl6eq 2672	. . . 4 |- ( ( _V \ A ) = A -> ( _V \ A ) = _V )
11	10, 6	eqtr3d 2658	. . 3 |- ( ( _V \ A ) = A -> _V = (/) )
12	11	necon3i 2826	. 2 |- ( _V =/= (/) -> ( _V \ A ) =/= A )
13	1, 12	ax-mp 5	1 |- ( _V \ A ) =/= A

Syntax hints:   = wceq 1483   =/= wne 2794   _Vcvv 3200   \ cdif 3571   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by: (None)