Theorem nelpr

Description: A set A not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023)

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

Proof

Step	Hyp	Ref	Expression
1		elprg	|- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
2	1	notbid	|- ( A e. V -> ( -. A e. { B , C } <-> -. ( A = B \/ A = C ) ) )
3		neanior	|- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4	2 3	bitr4di	|- ( A e. V -> ( -. A e. { B , C } <-> ( A =/= B /\ A =/= C ) ) )