Theorem eqoreldif

Description: An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020) (Proof shortened by JJ, 23-Jul-2021)

	Ref	Expression
Assertion	eqoreldif	|- ( B e. C -> ( A e. C <-> ( A = B \/ A e. ( C \ { B } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( A e. C /\ -. A = B ) -> A e. C )
2		elsni	|- ( A e. { B } -> A = B )
3	2	con3i	|- ( -. A = B -> -. A e. { B } )
4	3	adantl	|- ( ( A e. C /\ -. A = B ) -> -. A e. { B } )
5	1 4	eldifd	|- ( ( A e. C /\ -. A = B ) -> A e. ( C \ { B } ) )
6	5	ex	|- ( A e. C -> ( -. A = B -> A e. ( C \ { B } ) ) )
7	6	orrd	|- ( A e. C -> ( A = B \/ A e. ( C \ { B } ) ) )
8		eleq1a	|- ( B e. C -> ( A = B -> A e. C ) )
9		eldifi	|- ( A e. ( C \ { B } ) -> A e. C )
10	9	a1i	|- ( B e. C -> ( A e. ( C \ { B } ) -> A e. C ) )
11	8 10	jaod	|- ( B e. C -> ( ( A = B \/ A e. ( C \ { B } ) ) -> A e. C ) )
12	7 11	impbid2	|- ( B e. C -> ( A e. C <-> ( A = B \/ A e. ( C \ { B } ) ) ) )