Theorem opnoncon

Description: Law of contradiction for orthoposets. ( chocin analog.) (Contributed by NM, 13-Sep-2011)

	Ref	Expression
Hypotheses	opnoncon.b	|- B = ( Base ` K )
	opnoncon.o	|- ._|_ = ( oc ` K )
	opnoncon.m	|- ./\ = ( meet ` K )
	opnoncon.z	|- .0. = ( 0. ` K )
Assertion	opnoncon	|- ( ( K e. OP /\ X e. B ) -> ( X ./\ ( ._|_ ` X ) ) = .0. )

Proof

Step	Hyp	Ref	Expression
1		opnoncon.b	|- B = ( Base ` K )
2		opnoncon.o	|- ._|_ = ( oc ` K )
3		opnoncon.m	|- ./\ = ( meet ` K )
4		opnoncon.z	|- .0. = ( 0. ` K )
5		eqid	|- ( le ` K ) = ( le ` K )
6		eqid	|- ( join ` K ) = ( join ` K )
7		eqid	|- ( 1. ` K ) = ( 1. ` K )
8	1 5 2 6 3 4 7	oposlem	|- ( ( K e. OP /\ X e. B /\ X e. B ) -> ( ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X ( le ` K ) X -> ( ._|_ ` X ) ( le ` K ) ( ._|_ ` X ) ) ) /\ ( X ( join ` K ) ( ._|_ ` X ) ) = ( 1. ` K ) /\ ( X ./\ ( ._|_ ` X ) ) = .0. ) )
9	8	3anidm23	|- ( ( K e. OP /\ X e. B ) -> ( ( ( ._|_ ` X ) e. B /\ ( ._|_ ` ( ._|_ ` X ) ) = X /\ ( X ( le ` K ) X -> ( ._|_ ` X ) ( le ` K ) ( ._|_ ` X ) ) ) /\ ( X ( join ` K ) ( ._|_ ` X ) ) = ( 1. ` K ) /\ ( X ./\ ( ._|_ ` X ) ) = .0. ) )
10	9	simp3d	|- ( ( K e. OP /\ X e. B ) -> ( X ./\ ( ._|_ ` X ) ) = .0. )