Nuprl Lemma : setmem-intersectionset
∀s,x:coSet{i:l}. ((∃a:coSet{i:l}. (a ∈ s)) ⇒ ((x ∈ ⋂(s)) ⇐⇒ ∀z:coSet{i:l}. ((z ∈ s) ⇒ (x ∈ z))))
Proof
Definitions occuring in Statement :
intersectionset: ⋂(s),
setmem: (x ∈ s),
coSet: coSet{i:l},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
guard: {T},
cand: A c∧ B,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
allsetmem: ∀a∈A.P[a],
set-predicate: set-predicate{i:l}(s;a.P[a]),
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
intersectionset: ⋂(s),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
setmem-unionset,
seteq_weakening,
setmem_functionality,
allsetmem-iff,
exists_wf,
all_wf,
iff_wf,
sub-set_wf,
seteq_wf,
set-item_wf,
setmem_functionality_1,
coSet_wf,
setmem_wf,
allsetmem_wf,
unionset_wf,
setmem-sub-coset
Rules used in proof :
andLevelFunctionality,
because_Cache,
dependent_pairFormation,
functionEquality,
productEquality,
instantiate,
allLevelFunctionality,
promote_hyp,
levelHypothesis,
allFunctionality,
independent_functionElimination,
cumulativity,
setEquality,
rename,
setElimination,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
isectElimination,
dependent_functionElimination,
extract_by_obid,
introduction,
impliesFunctionality,
independent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
addLevel,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s,x:coSet\{i:l\}. ((\mexists{}a:coSet\{i:l\}. (a \mmember{} s)) {}\mRightarrow{} ((x \mmember{} \mcap{}(s)) \mLeftarrow{}{}\mRightarrow{} \mforall{}z:coSet\{i:l\}. ((z \mmember{} s) {}\mRightarrow{} (x \mmember{} z))))
Date html generated:
2018_07_29-AM-10_01_02
Last ObjectModification:
2018_07_18-PM-02_50_25
Theory : constructive!set!theory
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