Nuprl Lemma : l_member_settype
∀[T:Type]. ∀[x:T]. ∀[P:T ⟶ ℙ]. ∀[d:{i:T| P[i]} List]. x ∈ {x:T| P[x]} supposing (x ∈ d)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
true: True
Lemmas referenced :
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
member_wf,
subtype_rel_self,
subtype_rel_wf,
l_member_wf,
subtype_rel_list,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
hyp_replacement,
equalitySymmetry,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
Error :equalityIstype,
Error :inhabitedIsType,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
applyEquality,
instantiate,
universeEquality,
axiomEquality,
setEquality,
Error :setIsType,
Error :isectIsTypeImplies,
Error :functionIsType
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[d:\{i:T| P[i]\} List]. x \mmember{} \{x:T| P[x]\} supposing (x \mmember{} d)
Date html generated:
2019_06_20-PM-01_24_47
Last ObjectModification:
2018_12_07-PM-06_36_59
Theory : list_1
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