Nuprl Lemma : co-regext-transitive
∀a:coSet{i:l}. transitive-set(co-regext(a))
Proof
Definitions occuring in Statement :
co-regext: co-regext(a),
transitive-set: transitive-set(s),
coSet: coSet{i:l},
all: ∀x:A. B[x]
Definitions unfolded in proof :
Wsup: Wsup(a;b),
mk-set: f"(T),
pi1: fst(t),
set-dom: set-dom(s),
regextfun: regextfun(f;w),
ext-eq: A ≡ B,
uimplies: b supposing a,
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
top: Top,
uall: ∀[x:A]. B[x],
co-regext: co-regext(a),
mk-coset: mk-coset(T;f),
subtype_rel: A ⊆r B,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
seteq_wf,
item_mk_set_lemma,
coW_wf,
subtype_rel_weakening,
set-dom_wf,
coW-ext,
setmem-iff,
setsubset_wf,
all_wf,
transitive-set-iff,
coSet_wf,
setmem_wf,
regextfun_wf,
co-seteq-iff,
mk-coset_wf,
co-regext_wf,
setsubset-iff,
setmem-mk-coset,
coSet_subtype,
subtype_coSet
Rules used in proof :
functionExtensionality,
dependent_pairFormation,
independent_isectElimination,
productEquality,
functionEquality,
cumulativity,
lambdaEquality,
instantiate,
allFunctionality,
addLevel,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
isectElimination,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
hypothesis,
extract_by_obid,
introduction,
hypothesis_subsumption,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}a:coSet\{i:l\}. transitive-set(co-regext(a))
Date html generated:
2018_07_29-AM-10_08_01
Last ObjectModification:
2018_07_21-PM-04_36_52
Theory : constructive!set!theory
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