### Nuprl Lemma : linorder_le_neg

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  (Linorder(T;x,y.R[x;y]) `` (∀a,b:T.  (¬R[a;b] `⇐⇒` strict_part(x,y.R[x;y];b;a))))`

Proof

Definitions occuring in Statement :  linorder: `Linorder(T;x,y.R[x; y])` strict_part: `strict_part(x,y.R[x; y];a;b)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` not: `¬A` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` member: `t ∈ T` prop: `ℙ` so_apply: `x[s1;s2]` rev_implies: `P `` Q` not: `¬A` false: `False` so_lambda: `λ2x y.t[x; y]` strict_part: `strict_part(x,y.R[x; y];a;b)` linorder: `Linorder(T;x,y.R[x; y])` connex: `Connex(T;x,y.R[x; y])` order: `Order(T;x,y.R[x; y])` anti_sym: `AntiSym(T;x,y.R[x; y])` trans: `Trans(T;x,y.E[x; y])` refl: `Refl(T;x,y.E[x; y])` or: `P ∨ Q`
Lemmas referenced :  not_wf strict_part_wf linorder_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin applyEquality hypothesisEquality hypothesis independent_functionElimination voidElimination sqequalRule lambdaEquality Error :functionIsType,  Error :universeIsType,  Error :inhabitedIsType,  universeEquality productElimination dependent_functionElimination unionElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
(Linorder(T;x,y.R[x;y])  {}\mRightarrow{}  (\mforall{}a,b:T.    (\mneg{}R[a;b]  \mLeftarrow{}{}\mRightarrow{}  strict\_part(x,y.R[x;y];b;a))))

Date html generated: 2019_06_20-PM-00_29_59
Last ObjectModification: 2018_09_26-PM-00_04_58

Theory : rel_1

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