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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