Nuprl Lemma : existssetmem-implies
∀A:coSet{i:l}. ∀[P:{a:coSet{i:l}| (a ∈ A)} ⟶ ℙ]. (∃a∈A.P[a] ⇒ (∃a:coSet{i:l}. ((a ∈ A) ∧ P[a])))
Proof
Definitions occuring in Statement :
existssetmem: ∃a∈A.P[a],
setmem: (x ∈ s),
coSet: coSet{i:l},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
mk-coset: mk-coset(T;f),
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
pi2: snd(t),
pi1: fst(t),
set-dom: set-dom(s),
set-item: set-item(s;x),
existssetmem: ∃a∈A.P[a],
subtype_rel: A ⊆r B,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
coSet_wf,
existssetmem_wf,
subtype_rel_self,
mk-coset_wf,
setmem_wf,
setmem-coset,
coSet_subtype,
subtype_coSet
Rules used in proof :
functionEquality,
cumulativity,
setEquality,
lambdaEquality,
universeEquality,
instantiate,
dependent_set_memberEquality,
isectElimination,
productEquality,
independent_pairFormation,
dependent_functionElimination,
dependent_pairFormation,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
hypothesis,
extract_by_obid,
introduction,
cut,
hypothesis_subsumption,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}A:coSet\{i:l\}. \mforall{}[P:\{a:coSet\{i:l\}| (a \mmember{} A)\} {}\mrightarrow{} \mBbbP{}]. (\mexists{}a\mmember{}A.P[a] {}\mRightarrow{} (\mexists{}a:coSet\{i:l\}. ((a \mmember{} A) \mwedge{} P[a])))
Date html generated:
2018_07_29-AM-10_00_48
Last ObjectModification:
2018_07_18-PM-04_50_07
Theory : constructive!set!theory
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