Nuprl Lemma : l_member_subtype
∀[A,B:Type]. ∀L:A List. ∀x:A. (x ∈ L) ⇒ (x ∈ L) supposing A ⊆r B
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
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,
and: P ∧ Q
Lemmas referenced :
equal_wf,
less_than_wf,
length_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_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
sqequalHypSubstitution,
productElimination,
dependent_pairFormation,
hypothesisEquality,
promote_hyp,
independent_pairFormation,
equalitySymmetry,
applyEquality,
hyp_replacement,
applyLambdaEquality,
extract_by_obid,
isectElimination,
cumulativity,
productEquality,
setElimination,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}L:A List. \mforall{}x:A. (x \mmember{} L) {}\mRightarrow{} (x \mmember{} L) supposing A \msubseteq{}r B
Date html generated:
2017_04_17-AM-07_25_23
Last ObjectModification:
2017_02_27-PM-04_04_16
Theory : list_1
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