Nuprl Lemma : l_member-settype
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀L:{x:T| P[x]} List. ∀x:{x:T| P[x]} . ((x ∈ L) ⇐⇒ (x ∈ L))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
rev_implies: P ⇐ Q,
squash: ↓T,
so_lambda: λ2x.t[x]
Lemmas referenced :
equal_functionality_wrt_subtype_rel2,
less_than_wf,
length_wf,
equal_wf,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
l_member_wf,
subtype_rel_list,
set_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
cut,
hypothesis,
lambdaEquality,
setElimination,
rename,
setEquality,
cumulativity,
applyEquality,
functionExtensionality,
because_Cache,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
productEquality,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
universeEquality,
dependent_set_memberEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}L:\{x:T| P[x]\} List. \mforall{}x:\{x:T| P[x]\} . ((x \mmember{} L) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L))
Date html generated:
2017_04_17-AM-07_25_00
Last ObjectModification:
2017_02_27-PM-04_03_42
Theory : list_1
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