Nuprl Lemma : intersectionset_functionality
∀a,b:coSet{i:l}. (seteq(a;b) ⇒ seteq(⋂(a);⋂(b)))
Proof
Definitions occuring in Statement :
intersectionset: ⋂(s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
intersectionset: ⋂(s),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
set-predicate: set-predicate{i:l}(s;a.P[a]),
allsetmem: ∀a∈A.P[a],
cand: A c∧ B,
guard: {T}
Lemmas referenced :
co-seteq-iff,
intersectionset_wf,
setmem-sub-coset,
unionset_wf,
allsetmem_wf,
setmem_wf,
coSet_wf,
setmem_functionality_1,
set-item_wf,
set-dom_wf,
seteq_wf,
allsetmem-iff,
setmem_functionality,
seteq_weakening,
unionset_functionality,
seteq_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
independent_pairFormation,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
cumulativity,
because_Cache
Latex:
\mforall{}a,b:coSet\{i:l\}. (seteq(a;b) {}\mRightarrow{} seteq(\mcap{}(a);\mcap{}(b)))
Date html generated:
2019_10_31-AM-06_33_46
Last ObjectModification:
2018_08_31-AM-00_43_42
Theory : constructive!set!theory
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