Nuprl Lemma : refl_cl_is_order
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. trans(T;R) ⇒ order(T;Ro) supposing irrefl(T;R)
Proof
Definitions occuring in Statement :
refl_cl: Eo,
xxorder: order(T;R),
xxirrefl: irrefl(T;R),
xxtrans: trans(T;E),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
xxorder: order(T;R),
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,
and: P ∧ Q,
xxtrans: trans(T;E),
xxrefl: refl(T;E),
trans: Trans(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
refl_cl: Eo,
xxanti_sym: anti_sym(T;R),
anti_sym: AntiSym(T;x,y.R[x; y]),
or: P ∨ Q,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
refl_cl_wf,
xxtrans_wf,
xxirrefl_wf,
iff_weakening_equal,
equal_wf
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,
lambdaFormation,
independent_pairFormation,
lemma_by_obid,
because_Cache,
axiomEquality,
functionEquality,
cumulativity,
inlFormation,
unionElimination,
equalityTransitivity,
inrFormation,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. trans(T;R) {}\mRightarrow{} order(T;R\msupzero{}) supposing irrefl(T;R)
Date html generated:
2016_05_15-PM-00_01_47
Last ObjectModification:
2015_12_26-PM-11_26_05
Theory : gen_algebra_1
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