Nuprl Lemma : subset-co-regext-1
∀a:coSet{i:l}. (transitive-set(a) ⇒ (∀x:Set{i:l}. ((x ∈ a) ⇒ (x ∈ co-regext(a)))))
Proof
Definitions occuring in Statement :
co-regext: co-regext(a),
transitive-set: transitive-set(s),
Set: Set{i:l},
setmem: (x ∈ s),
coSet: coSet{i:l},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
mk-coset: mk-coset(T;f),
iff: P ⇐⇒ Q,
and: P ∧ Q,
set-item: set-item(s;x),
mk-set: f"(T),
set-dom: set-dom(s),
pi1: fst(t),
pi2: snd(t),
Wsup: Wsup(a;b),
exists: ∃x:A. B[x],
coset-relation: coSetRelation(R),
guard: {T},
mv-map: R:(A ⇒ B),
top: Top,
setsubset: (a ⊆ b),
allsetmem: ∀a∈A.P[a],
cand: A c∧ B,
onto-map: R:(A ─>> B),
rev_implies: P ⇐ Q
Lemmas referenced :
set-induction,
setmem_wf,
set-subtype-coSet,
co-regext_wf,
Set_wf,
subtype_coSet,
coSet_subtype,
setmem-iff,
mk-set_wf,
mk-coset_wf,
co-regext-lemma,
seteq_wf,
coSet_wf,
all_wf,
transitive-set_wf,
seteq_weakening,
equal_wf,
exists_wf,
set-dom_wf,
seteq_inversion,
seteq_transitivity,
transitive-set-iff,
setmem-coset,
setsubset_functionality,
setmem_functionality,
setmem-mk-set-sq,
dom_mk_set_lemma,
item_mk_set_lemma,
setsubset-iff,
setmem_functionality_1,
seteq-iff-setsubset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
cumulativity,
functionEquality,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
independent_functionElimination,
hypothesis_subsumption,
productElimination,
dependent_functionElimination,
rename,
universeEquality,
dependent_pairFormation,
instantiate,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
productEquality
Latex:
\mforall{}a:coSet\{i:l\}. (transitive-set(a) {}\mRightarrow{} (\mforall{}x:Set\{i:l\}. ((x \mmember{} a) {}\mRightarrow{} (x \mmember{} co-regext(a)))))
Date html generated:
2019_10_31-AM-06_34_15
Last ObjectModification:
2018_08_04-AM-10_31_35
Theory : constructive!set!theory
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