Nuprl Lemma : set-part-greatest
∀s:coSet{i:l}. ((set-part(s) ⊆ s) ∧ (∀x:Set{i:l}. ((x ⊆ s) ⇒ (x ⊆ set-part(s)))))
Proof
Definitions occuring in Statement :
setsubset: (a ⊆ b),
set-part: set-part(s),
Set: Set{i:l},
coSet: coSet{i:l},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
all: ∀x:A. B[x]
Lemmas referenced :
Set-isSet,
setsubset-iff2,
setmem_wf,
setmem-set-part,
set-part_wf,
setsubset-iff,
coSet_wf,
Set_wf,
set-subtype-coSet,
setsubset_wf
Rules used in proof :
because_Cache,
independent_functionElimination,
productElimination,
dependent_functionElimination,
sqequalRule,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s:coSet\{i:l\}. ((set-part(s) \msubseteq{} s) \mwedge{} (\mforall{}x:Set\{i:l\}. ((x \msubseteq{} s) {}\mRightarrow{} (x \msubseteq{} set-part(s)))))
Date html generated:
2018_07_29-AM-10_01_29
Last ObjectModification:
2018_07_25-PM-04_01_03
Theory : constructive!set!theory
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