Theorem nelprd

Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018)

	Ref	Expression
Hypotheses	nelprd.1	|- ( ph -> A =/= B )
	nelprd.2	|- ( ph -> A =/= C )
Assertion	nelprd	|- ( ph -> -. A e. { B , C } )

Proof

Step	Hyp	Ref	Expression
1		nelprd.1	|- ( ph -> A =/= B )
2		nelprd.2	|- ( ph -> A =/= C )
3		neanior	|- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4		elpri	|- ( A e. { B , C } -> ( A = B \/ A = C ) )
5	4	con3i	|- ( -. ( A = B \/ A = C ) -> -. A e. { B , C } )
6	3 5	sylbi	|- ( ( A =/= B /\ A =/= C ) -> -. A e. { B , C } )
7	1 2 6	syl2anc	|- ( ph -> -. A e. { B , C } )