Nuprl Lemma : setTC-least
∀a:Set{i:l}. ∀s:coSet{i:l}. ((a ⊆ s) ⇒ transitive-set(s) ⇒ (setTC(a) ⊆ s))
Proof
Definitions occuring in Statement :
transitive-set: transitive-set(s),
setsubset: (a ⊆ b),
setTC: setTC(a),
Set: Set{i:l},
coSet: coSet{i:l},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
top: Top,
guard: {T},
or: P ∨ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
setTC: Error :setTC,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
setTC_functionality,
coSet-seteq-Set,
seteq_inversion,
seteq_weakening,
setmem_functionality,
setmem-mk-set-sq,
setTC-set-function,
setmem-setunionfun,
coSet-mem-Set-implies-Set,
setunionfun_wf,
setmem-set-add,
transitive-set_wf,
transitive-set-iff,
setsubset_wf,
setsubset-iff,
set-subtype-coSet,
mk-set_wf,
Set_wf,
setmem_wf,
coSet_wf,
all_wf,
set-induction
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
voidEquality,
voidElimination,
isect_memberEquality,
unionElimination,
setEquality,
dependent_pairFormation,
independent_isectElimination,
rename,
setElimination,
functionExtensionality,
impliesLevelFunctionality,
allLevelFunctionality,
productElimination,
dependent_functionElimination,
impliesFunctionality,
allFunctionality,
addLevel,
universeEquality,
lambdaFormation,
independent_functionElimination,
because_Cache,
applyEquality,
hypothesisEquality,
cumulativity,
functionEquality,
hypothesis,
instantiate,
lambdaEquality,
sqequalRule,
thin,
isectElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}a:Set\{i:l\}. \mforall{}s:coSet\{i:l\}. ((a \msubseteq{} s) {}\mRightarrow{} transitive-set(s) {}\mRightarrow{} (setTC(a) \msubseteq{} s))
Date html generated:
2018_07_29-AM-10_03_42
Last ObjectModification:
2018_07_18-PM-08_56_51
Theory : constructive!set!theory
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