Nuprl Lemma : member-eq-is-equiv
∀[A,B:Type]. EquivRel(A;x,y.(x ∈ B) = (y ∈ B) ∈ Type) supposing respects-equality(A;B)
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uimplies: b supposing a,
respects-equality: respects-equality(S;T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
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],
sym: Sym(T;x,y.E[x; y]),
implies: P ⇒ Q,
trans: Trans(T;x,y.E[x; y]),
prop: ℙ
Lemmas referenced :
respects-equality_wf,
istype-universe,
equal-wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
Error :lambdaEquality_alt,
dependent_functionElimination,
hypothesisEquality,
axiomEquality,
hypothesis,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :universeIsType,
extract_by_obid,
isectElimination,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
instantiate,
universeEquality,
independent_pairFormation,
Error :lambdaFormation_alt,
because_Cache,
equalitySymmetry,
Error :equalityIstype,
equalityTransitivity,
independent_functionElimination
Latex:
\mforall{}[A,B:Type]. EquivRel(A;x,y.(x \mmember{} B) = (y \mmember{} B)) supposing respects-equality(A;B)
Date html generated:
2019_06_20-PM-00_30_12
Last ObjectModification:
2018_11_25-PM-06_18_55
Theory : rel_1
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