Nuprl Lemma : bag-member-single
∀[T:Type]. ∀[x,y:T]. uiff(x ↓∈ {y};x = y ∈ T)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
single-bag: {x},
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
bag-member: x ↓∈ bs,
squash: ↓T,
exists: ∃x:A. B[x],
single-bag: {x},
bag: bag(T),
quotient: x,y:A//B[x; y],
cand: A c∧ B,
permutation: permutation(T;L1;L2),
top: Top,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
l_member: (x ∈ l),
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
true: True,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
select: L[n],
cons: [a / b],
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
bag-member_wf,
single-bag_wf,
equal_wf,
member_wf,
list_wf,
cons_wf,
nil_wf,
permutation_wf,
length_wf,
length_of_cons_lemma,
length_of_nil_lemma,
permute_list_length,
subtype_base_sq,
int_subtype_base,
squash_wf,
true_wf,
select_wf,
le_wf,
less_than_wf,
false_wf,
length-singleton,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
iff_weakening_equal,
permute_list_select,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
int_seg_properties,
lelt_wf,
set_subtype_base,
non_neg_length,
length_wf_nat,
decidable__equal_int,
cons_member,
l_member_wf,
bag_wf,
list-subtype-bag
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
pertypeElimination,
productEquality,
applyLambdaEquality,
voidElimination,
voidEquality,
instantiate,
intEquality,
independent_isectElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
natural_numberEquality,
lambdaFormation,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
dependent_set_memberEquality,
functionExtensionality,
hyp_replacement,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[x,y:T]. uiff(x \mdownarrow{}\mmember{} \{y\};x = y)
Date html generated:
2017_10_01-AM-08_53_32
Last ObjectModification:
2017_07_26-PM-04_35_11
Theory : bags
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