Nuprl Lemma : quot_elim
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. (EquivRel(T;x,y.E[x;y]) ⇒ (∀a,b:T. (a = b ∈ (x,y:T//E[x;y]) ⇐⇒ ↓E[a;b])))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
prop: ℙ,
quotient: x,y:A//B[x; y],
cand: A c∧ B
Lemmas referenced :
quotient_wf,
subtype_quotient,
squash_wf,
istype-universe,
equiv_rel_wf,
subtype_rel_self,
quotient-member-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
Error :equalityIsType1,
Error :universeIsType,
extract_by_obid,
isectElimination,
Error :lambdaEquality_alt,
applyEquality,
Error :inhabitedIsType,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_pairEquality,
Error :functionIsTypeImplies,
axiomEquality,
Error :functionIsType,
universeEquality,
Error :isect_memberEquality_alt,
pertypeElimination,
Error :productIsType,
because_Cache,
instantiate,
independent_functionElimination,
lambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (EquivRel(T;x,y.E[x;y]) {}\mRightarrow{} (\mforall{}a,b:T. (a = b \mLeftarrow{}{}\mRightarrow{} \mdownarrow{}E[a;b])))
Date html generated:
2019_06_20-PM-00_32_10
Last ObjectModification:
2018_10_06-PM-03_56_26
Theory : quot_1
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