Nuprl Lemma : rel_le_refl_cl_sp

`∀[T:Type]. ∀[r:T ⟶ T ⟶ ℙ].  (dec_binrel(T;x,y:T. x = y ∈ T) `` r ≡>{T} (r\\00B8) supposing anti_sym(T;r))`

Proof

Definitions occuring in Statement :  s_part: `E\` refl_cl: `Eo` xxanti_sym: `anti_sym(T;R)` dec_binrel: `dec_binrel(T;r)` ab_binrel: `x,y:T. E[x; y]` binrel_le: `E ≡>{T} E'` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  s_part: `E\` refl_cl: `Eo` binrel_le: `E ≡>{T} E'` xxanti_sym: `anti_sym(T;R)` dec_binrel: `dec_binrel(T;r)` anti_sym: `AntiSym(T;x,y.R[x; y])` ab_binrel: `x,y:T. E[x; y]` uall: `∀[x:A]. B[x]` implies: `P `` Q` uimplies: `b supposing a` member: `t ∈ T` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` and: `P ∧ Q` cand: `A c∧ B` not: `¬A` false: `False` guard: `{T}`
Lemmas referenced :  all_wf equal_wf decidable_wf and_wf not_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation cut introduction sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality axiomEquality hypothesis applyEquality universeEquality because_Cache rename lemma_by_obid isectElimination functionEquality cumulativity unionElimination inlFormation inrFormation independent_pairFormation independent_functionElimination

Latex:
\mforall{}[T:Type].  \mforall{}[r:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    (dec\_binrel(T;x,y:T.  x  =  y)  {}\mRightarrow{}  r  \mequiv{}>\{T\}  (r\mbackslash{}\msupzero{})  supposing  anti\_sym(T;r))

Date html generated: 2016_05_15-PM-00_01_57
Last ObjectModification: 2015_12_26-PM-11_25_55

Theory : gen_algebra_1

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