Theorem compneOLD 38644

Description: Obsolete proof of compne 38643 as of 11-Nov-2021. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem compneOLD

Step	Hyp	Ref	Expression
1		vn0 3924	. 2 |- _V =/= (/)
2		ssun1 3776	. . . . . . . 8 |- _V C_ ( _V u. A )
3		ssv 3625	. . . . . . . 8 |- ( _V u. A ) C_ _V
4	2, 3	eqssi 3619	. . . . . . 7 |- _V = ( _V u. A )
5		undif1 4043	. . . . . . 7 |- ( ( _V \ A ) u. A ) = ( _V u. A )
6	4, 5	eqtr4i 2647	. . . . . 6 |- _V = ( ( _V \ A ) u. A )
7		uneq1 3760	. . . . . 6 |- ( ( _V \ A ) = A -> ( ( _V \ A ) u. A ) = ( A u. A ) )
8	6, 7	syl5eq 2668	. . . . 5 |- ( ( _V \ A ) = A -> _V = ( A u. A ) )
9		unidm 3756	. . . . 5 |- ( A u. A ) = A
10	8, 9	syl6eq 2672	. . . 4 |- ( ( _V \ A ) = A -> _V = A )
11		difabs 3892	. . . . . . 7 |- ( ( _V \ A ) \ A ) = ( _V \ A )
12		id 22	. . . . . . 7 |- ( ( _V \ A ) = A -> ( _V \ A ) = A )
13	11, 12	syl5req 2669	. . . . . 6 |- ( ( _V \ A ) = A -> A = ( ( _V \ A ) \ A ) )
14		difeq1 3721	. . . . . 6 |- ( ( _V \ A ) = A -> ( ( _V \ A ) \ A ) = ( A \ A ) )
15	13, 14	eqtrd 2656	. . . . 5 |- ( ( _V \ A ) = A -> A = ( A \ A ) )
16		difid 3948	. . . . 5 |- ( A \ A ) = (/)
17	15, 16	syl6eq 2672	. . . 4 |- ( ( _V \ A ) = A -> A = (/) )
18	10, 17	eqtrd 2656	. . 3 |- ( ( _V \ A ) = A -> _V = (/) )
19	18	necon3i 2826	. 2 |- ( _V =/= (/) -> ( _V \ A ) =/= A )
20	1, 19	ax-mp 5	1 |- ( _V \ A ) =/= A

Syntax hints:   = wceq 1483   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   (/)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)