Nuprl Lemma : l_member_in_subtype
∀[T:Type]. ∀x:T. ∀[P:T ⟶ ℙ]. ∀d:{i:T| P i} List. ((x ∈ d) ⇒ (x ∈ d))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
member: t ∈ T,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓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,
iff_weakening_equal,
equal_wf,
squash_wf,
true_wf,
istype-less_than,
length_wf,
l_member_wf,
subtype_rel_list,
subtype_rel_self,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
setEquality,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
sqequalRule,
setElimination,
rename,
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,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
Error :dependent_set_memberEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
Error :productIsType,
Error :equalityIstype,
applyLambdaEquality,
instantiate,
universeEquality,
Error :setIsType,
Error :functionIsType
Latex:
\mforall{}[T:Type]. \mforall{}x:T. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}d:\{i:T| P i\} List. ((x \mmember{} d) {}\mRightarrow{} (x \mmember{} d))
Date html generated:
2019_06_20-PM-01_24_51
Last ObjectModification:
2018_11_28-PM-03_37_19
Theory : list_1
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