Nuprl Lemma : sqle-mono-implies-equal
∀[T:Type]. ∀[x,y:Base]. (x = y ∈ T) supposing ((x ∈ T) and (x ≤ y)) supposing mono(T)
Proof
Definitions occuring in Statement :
mono: mono(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
base: Base,
universe: Type,
sqle: s ≤ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
mono: mono(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
is-above: is-above(T;a;z),
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ
Lemmas referenced :
istype-sqle,
istype-base,
mono_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
dependent_functionElimination,
thin,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
independent_functionElimination,
Error :dependent_pairFormation_alt,
because_Cache,
independent_pairFormation,
sqequalRule,
Error :productIsType,
Error :equalityIstype,
Error :inhabitedIsType,
sqequalBase,
extract_by_obid,
isectElimination,
Error :universeIsType,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x,y:Base]. (x = y) supposing ((x \mmember{} T) and (x \mleq{} y)) supposing mono(T)
Date html generated:
2019_06_20-PM-00_28_27
Last ObjectModification:
2019_01_20-PM-03_19_49
Theory : subtype_1
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