Theorem prprc1

Description: A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993)

	Ref	Expression
Assertion	prprc1	|- ( -. A e. _V -> { A , B } = { B } )

Proof

Step	Hyp	Ref	Expression
1		snprc	|- ( -. A e. _V <-> { A } = (/) )
2		uneq1	|- ( { A } = (/) -> ( { A } u. { B } ) = ( (/) u. { B } ) )
3		df-pr	|- { A , B } = ( { A } u. { B } )
4		uncom	|- ( (/) u. { B } ) = ( { B } u. (/) )
5		un0	|- ( { B } u. (/) ) = { B }
6	4 5	eqtr2i	|- { B } = ( (/) u. { B } )
7	2 3 6	3eqtr4g	|- ( { A } = (/) -> { A , B } = { B } )
8	1 7	sylbi	|- ( -. A e. _V -> { A , B } = { B } )