Nuprl Lemma : sp_refl_cl_cancel
∀[T:Type]. ∀[r:T ⟶ T ⟶ ℙ]. ((ro\) <≡>{T} r) 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_eqv: 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 :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
xxirrefl: irrefl(T;R),
irrefl: Irrefl(T;x,y.E[x; y]),
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
xxst_anti_sym: st_anti_sym(T;R),
st_anti_sym: StAntiSym(T;x,y.R[x; y]),
all: ∀x:A. B[x],
and: P ∧ Q
Lemmas referenced :
and_wf,
binrel_le_antisymmetry,
s_part_wf,
refl_cl_wf,
sp_refl_cl_le_rel,
rel_le_sp_refl_cl,
xxst_anti_sym_wf,
xxirrefl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
applyEquality,
hypothesis,
universeEquality,
rename,
lemma_by_obid,
independent_functionElimination,
because_Cache,
independent_isectElimination,
functionEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[r:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((r\msupzero{}\mbackslash{}) <\mequiv{}>\{T\} r) supposing (st\_anti\_sym(T;r) and irrefl(T;r))
Date html generated:
2016_05_15-PM-00_02_10
Last ObjectModification:
2015_12_26-PM-11_25_37
Theory : gen_algebra_1
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