Theorem cvbr2

Description: Binary relation expressing B covers A . Definition of covers in Kalmbach p. 15. (Contributed by NM, 9-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvbr2	|- ( ( A e. CH /\ B e. CH ) -> ( A ( A C. B /\ A. x e. CH ( ( A C. x /\ x C_ B ) -> x = B ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cvbr	|- ( ( A e. CH /\ B e. CH ) -> ( A ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) ) )
2		iman	|- ( ( ( A C. x /\ x C_ B ) -> x = B ) <-> -. ( ( A C. x /\ x C_ B ) /\ -. x = B ) )
3		anass	|- ( ( ( A C. x /\ x C_ B ) /\ -. x = B ) <-> ( A C. x /\ ( x C_ B /\ -. x = B ) ) )
4		dfpss2	|- ( x C. B <-> ( x C_ B /\ -. x = B ) )
5	4	anbi2i	|- ( ( A C. x /\ x C. B ) <-> ( A C. x /\ ( x C_ B /\ -. x = B ) ) )
6	3 5	bitr4i	|- ( ( ( A C. x /\ x C_ B ) /\ -. x = B ) <-> ( A C. x /\ x C. B ) )
7	2 6	xchbinx	|- ( ( ( A C. x /\ x C_ B ) -> x = B ) <-> -. ( A C. x /\ x C. B ) )
8	7	ralbii	|- ( A. x e. CH ( ( A C. x /\ x C_ B ) -> x = B ) <-> A. x e. CH -. ( A C. x /\ x C. B ) )
9		ralnex	|- ( A. x e. CH -. ( A C. x /\ x C. B ) <-> -. E. x e. CH ( A C. x /\ x C. B ) )
10	8 9	bitri	|- ( A. x e. CH ( ( A C. x /\ x C_ B ) -> x = B ) <-> -. E. x e. CH ( A C. x /\ x C. B ) )
11	10	anbi2i	|- ( ( A C. B /\ A. x e. CH ( ( A C. x /\ x C_ B ) -> x = B ) ) <-> ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) )
12	1 11	bitr4di	|- ( ( A e. CH /\ B e. CH ) -> ( A ( A C. B /\ A. x e. CH ( ( A C. x /\ x C_ B ) -> x = B ) ) ) )