Nuprl Lemma : setTC-induction
∀[P:Set{i:l} ⟶ ℙ']. ((∀a:Set{i:l}. ((∀x:Set{i:l}. ((x ∈ setTC(a)) ⇒ P[x])) ⇒ P[a])) ⇒ (∀s:Set{i:l}. P[s]))
Proof
Definitions occuring in Statement :
setTC: setTC(a),
Set: Set{i:l},
setmem: (x ∈ s),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
guard: {T},
top: Top,
or: P ∨ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
setTC: Error :setTC,
all: ∀x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coSet-seteq-Set,
setTC-set-function,
setmem-setunionfun,
setTC_functionality,
seteq_weakening,
setmem_functionality,
setmem-mk-set-sq,
coSet-mem-Set-implies-Set,
coSet_wf,
setunionfun_wf,
setmem-set-add,
mk-set_wf,
set-induction,
set-subtype-coSet,
setmem_wf,
Set_wf,
all_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
voidEquality,
voidElimination,
isect_memberEquality,
unionElimination,
productElimination,
setEquality,
dependent_pairFormation,
independent_isectElimination,
rename,
setElimination,
functionExtensionality,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
universeEquality,
applyEquality,
hypothesisEquality,
cumulativity,
functionEquality,
lambdaEquality,
sqequalRule,
hypothesis,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
instantiate,
thin,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[P:Set\{i:l\} {}\mrightarrow{} \mBbbP{}']
((\mforall{}a:Set\{i:l\}. ((\mforall{}x:Set\{i:l\}. ((x \mmember{} setTC(a)) {}\mRightarrow{} P[x])) {}\mRightarrow{} P[a])) {}\mRightarrow{} (\mforall{}s:Set\{i:l\}. P[s]))
Date html generated:
2018_07_29-AM-10_03_50
Last ObjectModification:
2018_07_18-PM-04_57_59
Theory : constructive!set!theory
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