Nuprl Lemma : quotient-member-eq
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. ∀x,y:T. (E[x;y] ⇒ (x = y ∈ (x,y:T//E[x;y]))) supposing EquivRel(T;x,y.E[x;y])
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
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,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
quotient: x,y:A//B[x; y],
and: P ∧ Q,
cand: A c∧ B,
squash: ↓T,
guard: {T},
true: True,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
subtype_rel_self,
equiv_rel_wf,
istype-universe,
quotient_wf,
equal_wf,
squash_wf,
true_wf,
subtype_quotient,
equal_functionality_wrt_subtype_rel2,
equal-wf-base-T,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
hypothesis,
Error :universeIsType,
applyEquality,
hypothesisEquality,
thin,
sqequalRule,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
universeEquality,
Error :inhabitedIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :functionIsType,
because_Cache,
pointwiseFunctionality,
pertypeMemberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_pairFormation,
imageElimination,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
Error :equalityIsType3,
applyLambdaEquality,
setElimination,
rename,
productElimination,
Error :equalityIsType1,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:T. (E[x;y] {}\mRightarrow{} (x = y)) supposing EquivRel(T;x,y.E[x;y])
Date html generated:
2019_06_20-PM-00_32_07
Last ObjectModification:
2018_11_24-AM-09_34_59
Theory : quot_1
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