Nuprl Lemma : set-induction-1
∀[P:Set{i:l} ⟶ ℙ']. ((∀T:Type. ∀f:T ⟶ Set{i:l}. ((∀t:T. P[f[t]]) ⇒ P[f"(T)])) ⇒ (∀s:Set{i:l}. P[s]))
Proof
Definitions occuring in Statement :
mk-set: f"(T),
Set: Set{i:l},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
prop: ℙ,
Set: Set{i:l},
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
mk-set: f"(T),
guard: {T}
Lemmas referenced :
mk-set_wf,
all_wf,
Set_wf,
W-induction
Rules used in proof :
applyEquality,
functionEquality,
hypothesis,
hypothesisEquality,
cumulativity,
lambdaEquality,
sqequalRule,
universeEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
instantiate,
thin,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[P:Set\{i:l\} {}\mrightarrow{} \mBbbP{}']
((\mforall{}T:Type. \mforall{}f:T {}\mrightarrow{} Set\{i:l\}. ((\mforall{}t:T. P[f[t]]) {}\mRightarrow{} P[f"(T)])) {}\mRightarrow{} (\mforall{}s:Set\{i:l\}. P[s]))
Date html generated:
2018_05_22-PM-09_47_52
Last ObjectModification:
2018_05_16-PM-03_19_22
Theory : constructive!set!theory
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