Theorem cvnsym

Description: The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvnsym	|- ( ( A e. CH /\ B e. CH ) -> ( A -. B

Proof

Step	Hyp	Ref	Expression
1		cvpss	|- ( ( A e. CH /\ B e. CH ) -> ( A A C. B ) )
2		cvpss	|- ( ( B e. CH /\ A e. CH ) -> ( B B C. A ) )
3	2	ancoms	|- ( ( A e. CH /\ B e. CH ) -> ( B B C. A ) )
4		pssn2lp	|- -. ( B C. A /\ A C. B )
5	4	imnani	|- ( B C. A -> -. A C. B )
6	3 5	syl6	|- ( ( A e. CH /\ B e. CH ) -> ( B -. A C. B ) )
7	6	con2d	|- ( ( A e. CH /\ B e. CH ) -> ( A C. B -> -. B
8	1 7	syld	|- ( ( A e. CH /\ B e. CH ) -> ( A -. B