Theorem nelbr 41291

Description: The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.)

Assertion

Ref	Expression
nelbr	|- ( ( A e. V /\ B e. W ) -> ( A _ e// B <-> -. A e. B ) )

Proof of Theorem nelbr

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

Step	Hyp	Ref	Expression
1		eleq12 2691	. . 3 |- ( ( x = A /\ y = B ) -> ( x e. y <-> A e. B ) )
2	1	notbid 308	. 2 |- ( ( x = A /\ y = B ) -> ( -. x e. y <-> -. A e. B ) )
3		df-nelbr 41289	. 2 |- _ e// = { <. x , y >. | -. x e. y }
4	2, 3	brabga 4989	1 |- ( ( A e. V /\ B e. W ) -> ( A _ e// B <-> -. A e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   _ 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-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-nelbr 41289

This theorem is referenced by:  nelbrim  41292  nelbrnel  41293