Nuprl Lemma : l_member_in_subtype2
∀[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,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
l_member-settype,
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,
set_wf,
and_wf,
equal_wf,
l_member_wf,
subtype_rel_list,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
dependent_functionElimination,
productElimination,
setEquality,
because_Cache,
hypothesis,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalitySymmetry,
dependent_set_memberEquality,
hyp_replacement,
Error :applyLambdaEquality,
equalityTransitivity,
independent_functionElimination,
universeEquality,
functionEquality
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:
2016_10_21-AM-10_03_27
Last ObjectModification:
2016_07_12-AM-05_24_17
Theory : list_1
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