Theorem prnesn

Description: A proper unordered pair is not a (proper or improper) singleton. (Contributed by AV, 13-Jun-2022)

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

Proof

Step	Hyp	Ref	Expression
1		eqtr3	|- ( ( A = C /\ B = C ) -> A = B )
2	1	necon3ai	|- ( A =/= B -> -. ( A = C /\ B = C ) )
3	2	3ad2ant3	|- ( ( A e. V /\ B e. W /\ A =/= B ) -> -. ( A = C /\ B = C ) )
4		simp1	|- ( ( A e. V /\ B e. W /\ A =/= B ) -> A e. V )
5		simp2	|- ( ( A e. V /\ B e. W /\ A =/= B ) -> B e. W )
6	4 5	preqsnd	|- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } = { C } <-> ( A = C /\ B = C ) ) )
7	6	necon3abid	|- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } =/= { C } <-> -. ( A = C /\ B = C ) ) )
8	3 7	mpbird	|- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } =/= { C } )