Nuprl Lemma : extend-type-equiv
∀[T:Type]. EquivRel(Base;x,y.(x ∈ T ⇐⇒ y ∈ T) ∧ ((x ∈ T) ⇒ (x = y ∈ T)))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
cand: A c∧ B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
istype-base,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
equalityTransitivity,
hypothesis,
equalitySymmetry,
sqequalRule,
Error :equalityIstype,
Error :universeIsType,
hypothesisEquality,
sqequalBase,
because_Cache,
productElimination,
thin,
extract_by_obid,
independent_functionElimination,
Error :productIsType,
Error :functionIsType,
independent_pairEquality,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
instantiate,
isectElimination,
universeEquality
Latex:
\mforall{}[T:Type]. EquivRel(Base;x,y.(x \mmember{} T \mLeftarrow{}{}\mRightarrow{} y \mmember{} T) \mwedge{} ((x \mmember{} T) {}\mRightarrow{} (x = y)))
Date html generated:
2019_06_20-PM-00_33_20
Last ObjectModification:
2018_11_25-PM-06_36_28
Theory : quot_1
Home
Index