Nuprl Lemma : inductive-set-property
∀[R:Set{i:l} ⟶ Set{i:l} ⟶ ℙ']. ∀bdd:Bounded(x,a.R[x;a]). inductively-defined{i:l}(x,a.R[x;a];inductive-set(bdd))
Proof
Definitions occuring in Statement :
inductive-set: inductive-set(bdd),
inductively-defined: inductively-defined{i:l}(x,a.R[x; a];s),
bounded-relation: Bounded(x,a.R[x; a]),
Set: Set{i:l},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
bounded-relation: Bounded(x,a.R[x; a]),
and: P ∧ Q,
exists: ∃x:A. B[x],
inductive-set: inductive-set(bdd),
spreadn: spread4,
so_lambda: λ2x y.t[x; y],
member: t ∈ T,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi1: fst(t),
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
least-closed-set-inductively-defined,
Set_wf,
closure-set_wf,
exists_wf,
all_wf,
iff_wf,
setmem_wf,
pi1_wf,
equal_wf,
bounded-relation_wf,
closure-set-property,
subtype_rel_self,
setsubset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
functionExtensionality,
instantiate,
cumulativity,
dependent_pairEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionEquality,
universeEquality,
addLevel,
allFunctionality,
impliesFunctionality,
independent_pairFormation,
because_Cache,
levelHypothesis,
existsFunctionality,
andLevelFunctionality,
productEquality,
promote_hyp
Latex:
\mforall{}[R:Set\{i:l\} {}\mrightarrow{} Set\{i:l\} {}\mrightarrow{} \mBbbP{}']
\mforall{}bdd:Bounded(x,a.R[x;a]). inductively-defined\{i:l\}(x,a.R[x;a];inductive-set(bdd))
Date html generated:
2018_07_29-AM-10_10_22
Last ObjectModification:
2018_05_30-PM-06_48_09
Theory : constructive!set!theory
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