Nuprl Lemma : rel-immediate-property
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(sum_of_torder(T;R) ⇒ (∀x,y,x',y':T. ((R x y) ⇒ (R! y' y) ⇒ ((R x y') ∨ (x = y' ∈ T)))))
Proof
Definitions occuring in Statement :
sum_of_torder: sum_of_torder(T;R),
rel-immediate: R!,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
member: t ∈ T,
sum_of_torder: sum_of_torder(T;R),
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
rel-immediate: R!,
and: P ∧ Q,
or: P ∨ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
not: ¬A,
false: False
Lemmas referenced :
subtype_rel_self,
rel-immediate_wf,
sum_of_torder_wf
Rules used in proof :
universeEquality,
cumulativity,
functionEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
lemma_by_obid,
cut,
applyEquality,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalRule,
productElimination,
dependent_functionElimination,
independent_functionElimination,
inlFormation_alt,
independent_pairFormation,
productIsType,
universeIsType,
instantiate,
introduction,
extract_by_obid,
because_Cache,
unionElimination,
equalityIstype,
inhabitedIsType,
inrFormation_alt,
voidElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(sum\_of\_torder(T;R) {}\mRightarrow{} (\mforall{}x,y,x',y':T. ((R x y) {}\mRightarrow{} (R! y' y) {}\mRightarrow{} ((R x y') \mvee{} (x = y')))))
Date html generated:
2020_05_20-AM-08_10_11
Last ObjectModification:
2020_01_17-PM-05_50_27
Theory : general
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