### Nuprl Lemma : refl_cl_sp_cancel

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

Proof

Definitions occuring in Statement :  s_part: `E\` refl_cl: `Eo` xxanti_sym: `anti_sym(T;R)` xxrefl: `refl(T;E)` dec_binrel: `dec_binrel(T;r)` ab_binrel: `x,y:T. E[x; y]` binrel_eqv: `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 :  uall: `∀[x:A]. B[x]` implies: `P `` Q` uimplies: `b supposing a` member: `t ∈ T` xxanti_sym: `anti_sym(T;R)` anti_sym: `AntiSym(T;x,y.R[x; y])` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]`
Lemmas referenced :  binrel_le_antisymmetry refl_cl_wf s_part_wf xxanti_sym_wf xxrefl_wf dec_binrel_wf ab_binrel_wf equal_wf refl_cl_sp_le_rel rel_le_refl_cl_sp
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality axiomEquality hypothesis applyEquality universeEquality because_Cache rename lemma_by_obid isectElimination independent_functionElimination functionEquality cumulativity independent_isectElimination

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

Date html generated: 2016_05_15-PM-00_02_02
Last ObjectModification: 2015_12_26-PM-11_25_46

Theory : gen_algebra_1

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