Nuprl Lemma : coSet-seteq-Set
∀[s:Set{i:l}]. ∀[z:coSet{i:l}]. z ∈ Set{i:l} supposing seteq(z;s)
Proof
Definitions occuring in Statement :
Set: Set{i:l},
seteq: seteq(s1;s2),
coSet: coSet{i:l},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Set: Set{i:l},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
implies: P ⇒ Q,
pcw-pp-barred: Barred(pp),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
cw-step: cw-step(A;a.B[a]),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
spreadn: spread3,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
isr: isr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
ext-eq: A ≡ B,
unit: Unit,
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ext-family: F ≡ G,
pi1: fst(t),
nat_plus: ℕ+,
W-rel: W-rel(A;a.B[a];w),
param-W-rel: param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w),
pcw-steprel: StepRel(s1;s2),
pi2: snd(t),
isl: isl(x),
pcw-step-agree: StepAgree(s;p1;w),
cand: A c∧ B,
guard: {T},
Wsup: Wsup(a;b),
sq_type: SQType(T),
le: A ≤ B,
sq_stable: SqStable(P),
mk-set: f"(T),
iff: P ⇐⇒ Q,
set-dom: set-dom(s),
set-item: set-item(s;x),
mk-coset: mk-coset(T;f),
rev_implies: P ⇐ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
seteq_wf,
set-subtype-coSet,
coSet_wf,
Set_wf,
W-elimination-facts,
subtype_rel_self,
int_seg_wf,
subtract_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
lelt_wf,
top_wf,
less_than_wf,
false_wf,
true_wf,
equal_wf,
add-subtract-cancel,
itermAdd_wf,
int_term_value_add_lemma,
W-ext,
param-co-W-ext,
unit_wf2,
it_wf,
param-co-W_wf,
pcw-steprel_wf,
subtype_rel_dep_function,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_function,
int_seg_subtype,
sq_stable__le,
mk-set_wf,
setmem_wf,
co-seteq-iff,
setmem-iff,
subtype_coSet,
coSet_subtype,
mk-coset_wf,
seteq-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
applyEquality,
instantiate,
dependent_functionElimination,
universeEquality,
lambdaEquality,
cumulativity,
productElimination,
strong_bar_Induction,
independent_functionElimination,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
lambdaFormation,
lessCases,
axiomSqEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
int_eqReduceTrueSq,
promote_hyp,
hypothesis_subsumption,
equalityElimination,
dependent_pairEquality,
inlEquality,
unionEquality,
productEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[s:Set\{i:l\}]. \mforall{}[z:coSet\{i:l\}]. z \mmember{} Set\{i:l\} supposing seteq(z;s)
Date html generated:
2019_10_31-AM-06_33_08
Last ObjectModification:
2018_08_23-AM-11_49_54
Theory : constructive!set!theory
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