Nuprl Lemma : setmem-unionset
∀s,x:coSet{i:l}. ((x ∈ ⋃(s)) ⇐⇒ ∃a:coSet{i:l}. ((a ∈ s) ∧ (x ∈ a)))
Proof
Definitions occuring in Statement :
unionset: ⋃(s),
setmem: (x ∈ s),
coSet: coSet{i:l},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
unionset: ⋃(s),
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
set-function: set-function{i:l}(s; x.f[x]),
all: ∀x:A. B[x]
Lemmas referenced :
iff_wf,
setunionfun_wf,
setmem-setunionfun,
exists_wf,
coSet_wf,
setmem_wf,
seteq_wf
Rules used in proof :
independent_functionElimination,
setEquality,
rename,
setElimination,
dependent_functionElimination,
impliesFunctionality,
productElimination,
addLevel,
cumulativity,
productEquality,
lambdaEquality,
sqequalRule,
instantiate,
independent_pairFormation,
because_Cache,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s,x:coSet\{i:l\}. ((x \mmember{} \mcup{}(s)) \mLeftarrow{}{}\mRightarrow{} \mexists{}a:coSet\{i:l\}. ((a \mmember{} s) \mwedge{} (x \mmember{} a)))
Date html generated:
2018_07_29-AM-09_53_02
Last ObjectModification:
2018_07_18-PM-02_46_03
Theory : constructive!set!theory
Home
Index