Nuprl Lemma : extend-type-property
∀[T:Type]. ((T ⊆r (T)+) ∧ respects-equality((T)+;T) ∧ (∀X:Type. (respects-equality(X;T) 
⇒ (X ⊆r (T)+))))
Proof
Definitions occuring in Statement : 
extend-type: (T)+
, 
subtype_rel: A ⊆r B
, 
respects-equality: respects-equality(S;T)
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
and: P ∧ Q
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
extend-type: (T)+
, 
so_lambda: λ2x y.t[x; y]
, 
prop: ℙ
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
so_apply: x[s1;s2]
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
cand: A c∧ B
, 
respects-equality: respects-equality(S;T)
, 
quotient: x,y:A//B[x; y]
Lemmas referenced : 
istype-universe, 
extend-type_wf, 
quotient-member-eq, 
base_wf, 
iff_wf, 
equal-wf-base, 
equal-wf-T-base, 
extend-type-equiv, 
istype-base, 
respects-equality_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
independent_pairFormation, 
cut, 
instantiate, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
universeEquality, 
hypothesis, 
Error :lambdaEquality_alt, 
Error :universeIsType, 
hypothesisEquality, 
pointwiseFunctionalityForEquality, 
sqequalRule, 
productEquality, 
because_Cache, 
functionEquality, 
Error :inhabitedIsType, 
independent_isectElimination, 
dependent_functionElimination, 
independent_functionElimination, 
Error :lambdaFormation_alt, 
equalityTransitivity, 
equalitySymmetry, 
Error :equalityIstype, 
sqequalBase, 
productElimination, 
pertypeElimination, 
promote_hyp, 
Error :productIsType, 
Error :functionIsType, 
axiomEquality
Latex:
\mforall{}[T:Type]
    ((T  \msubseteq{}r  (T)+)  \mwedge{}  respects-equality((T)+;T)  \mwedge{}  (\mforall{}X:Type.  (respects-equality(X;T)  {}\mRightarrow{}  (X  \msubseteq{}r  (T)+))))
Date html generated:
2019_06_20-PM-00_33_29
Last ObjectModification:
2018_11_25-PM-06_55_25
Theory : quot_1
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