Theorem nelbOLDOLD

Description: Obsolete version of nelb as of 23-Jan-2024. (Contributed by Thierry Arnoux, 20-Nov-2023) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	nelbOLDOLD	|- ( -. A e. B <-> A. x e. B x =/= A )

Proof

Step	Hyp	Ref	Expression
1		df-ral	|- ( A. x e. B -. x = A <-> A. x ( x e. B -> -. x = A ) )
2		df-ne	|- ( x =/= A <-> -. x = A )
3	2	ralbii	|- ( A. x e. B x =/= A <-> A. x e. B -. x = A )
4		dfclel	|- ( A e. B <-> E. x ( x = A /\ x e. B ) )
5	4	notbii	|- ( -. A e. B <-> -. E. x ( x = A /\ x e. B ) )
6		alnex	|- ( A. x -. ( x = A /\ x e. B ) <-> -. E. x ( x = A /\ x e. B ) )
7		imnan	|- ( ( x e. B -> -. x = A ) <-> -. ( x e. B /\ x = A ) )
8		ancom	|- ( ( x e. B /\ x = A ) <-> ( x = A /\ x e. B ) )
9	8	notbii	|- ( -. ( x e. B /\ x = A ) <-> -. ( x = A /\ x e. B ) )
10	7 9	bitr2i	|- ( -. ( x = A /\ x e. B ) <-> ( x e. B -> -. x = A ) )
11	10	albii	|- ( A. x -. ( x = A /\ x e. B ) <-> A. x ( x e. B -> -. x = A ) )
12	5 6 11	3bitr2i	|- ( -. A e. B <-> A. x ( x e. B -> -. x = A ) )
13	1 3 12	3bitr4ri	|- ( -. A e. B <-> A. x e. B x =/= A )