Theorem dfnelbr2 41290

Description: Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.)

Assertion

Ref	Expression
dfnelbr2	|- _ e// = ( ( _V X. _V ) \ _E )

Proof of Theorem dfnelbr2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difopab 5253	. 2 |- ( { <. x , y >. | ( x e. _V /\ y e. _V ) } \ { <. x , y >. | x e. y } ) = { <. x , y >. | ( ( x e. _V /\ y e. _V ) /\ -. x e. y ) }
2		df-xp 5120	. . 3 |- ( _V X. _V ) = { <. x , y >. | ( x e. _V /\ y e. _V ) }
3		df-eprel 5029	. . 3 |- _E = { <. x , y >. | x e. y }
4	2, 3	difeq12i 3726	. 2 |- ( ( _V X. _V ) \ _E ) = ( { <. x , y >. | ( x e. _V /\ y e. _V ) } \ { <. x , y >. | x e. y } )
5		df-nelbr 41289	. . 3 |- _ e// = { <. x , y >. | -. x e. y }
6		vex 3203	. . . . . 6 |- x e. _V
7		vex 3203	. . . . . 6 |- y e. _V
8	6, 7	pm3.2i 471	. . . . 5 |- ( x e. _V /\ y e. _V )
9	8	biantrur 527	. . . 4 |- ( -. x e. y <-> ( ( x e. _V /\ y e. _V ) /\ -. x e. y ) )
10	9	opabbii 4717	. . 3 |- { <. x , y >. | -. x e. y } = { <. x , y >. | ( ( x e. _V /\ y e. _V ) /\ -. x e. y ) }
11	5, 10	eqtri 2644	. 2 |- _ e// = { <. x , y >. | ( ( x e. _V /\ y e. _V ) /\ -. x e. y ) }
12	1, 4, 11	3eqtr4ri 2655	1 |- _ e// = ( ( _V X. _V ) \ _E )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   {copab 4712   _E cep 5028   X. cxp 5112   _ e// cnelbr 41288

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-eprel 5029  df-xp 5120  df-rel 5121  df-nelbr 41289

This theorem is referenced by: (None)