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Metamath Proof Explorer

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 ) ) ) )