Nuprl Lemma : rel_le_sp_refl_cl

[T:Type]. ∀[r:T ⟶ T ⟶ ℙ].  (r ≡>{T} (ro\)) supposing (st_anti_sym(T;r) and irrefl(T;r))


Definitions occuring in Statement :  s_part: E\ refl_cl: Eo xxst_anti_sym: st_anti_sym(T;R) xxirrefl: irrefl(T;R) binrel_le: E ≡>{T} E' uimplies: supposing a uall: [x:A]. B[x] prop: function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  refl_cl: Eo s_part: E\ binrel_le: E ≡>{T} E' xxst_anti_sym: st_anti_sym(T;R) xxirrefl: irrefl(T;R) st_anti_sym: StAntiSym(T;x,y.R[x; y]) irrefl: Irrefl(T;x,y.E[x; y]) uall: [x:A]. B[x] uimplies: supposing a member: t ∈ T not: ¬A implies:  Q false: False subtype_rel: A ⊆B prop: all: x:A. B[x] and: P ∧ Q so_lambda: λ2x.t[x] so_apply: x[s] or: P ∨ Q guard: {T} iff: ⇐⇒ Q cand: c∧ B
Lemmas referenced :  and_wf all_wf not_wf uall_wf equal_wf or_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation cut introduction sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality lambdaEquality dependent_functionElimination voidElimination applyEquality hypothesis universeEquality rename lemma_by_obid lambdaFormation functionEquality cumulativity independent_pairFormation inrFormation unionElimination equalityTransitivity equalitySymmetry independent_isectElimination productElimination independent_functionElimination

\mforall{}[T:Type].  \mforall{}[r:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    (r  \mequiv{}>\{T\}  (r\msupzero{}\mbackslash{}))  supposing  (st\_anti\_sym(T;r)  and  irrefl(T;r))

Date html generated: 2016_05_15-PM-00_02_05
Last ObjectModification: 2015_12_26-PM-11_25_50

Theory : gen_algebra_1

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