This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cvrnrefN

Description: The covers relation is not reflexive. ( cvnref analog.) (Contributed by NM, 1-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cvrne.b 
|- B = ( Base ` K )
cvrne.c 
|- C = (
Assertion cvrnrefN 
|- ( ( K e. A /\ X e. B ) -> -. X C X )

Proof

Step Hyp Ref Expression
1 cvrne.b 
 |-  B = ( Base ` K )
2 cvrne.c 
 |-  C = (
3 eqid 
 |-  X = X
4 simpll 
 |-  ( ( ( K e. A /\ X e. B ) /\ X C X ) -> K e. A )
5 simplr 
 |-  ( ( ( K e. A /\ X e. B ) /\ X C X ) -> X e. B )
6 simpr 
 |-  ( ( ( K e. A /\ X e. B ) /\ X C X ) -> X C X )
7 1 2 cvrne 
 |-  ( ( ( K e. A /\ X e. B /\ X e. B ) /\ X C X ) -> X =/= X )
8 4 5 5 6 7 syl31anc 
 |-  ( ( ( K e. A /\ X e. B ) /\ X C X ) -> X =/= X )
9 8 ex 
 |-  ( ( K e. A /\ X e. B ) -> ( X C X -> X =/= X ) )
10 9 necon2bd 
 |-  ( ( K e. A /\ X e. B ) -> ( X = X -> -. X C X ) )
11 3 10 mpi 
 |-  ( ( K e. A /\ X e. B ) -> -. X C X )