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))`

Proof

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: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` 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: `b supposing a` member: `t ∈ T` not: `¬A` implies: `P `` Q` false: `False` subtype_rel: `A ⊆r B` prop: `ℙ` all: `∀x:A. B[x]` and: `P ∧ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` or: `P ∨ Q` guard: `{T}` iff: `P `⇐⇒` Q` cand: `A 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

Latex:
\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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