Nuprl Lemma : allsetmem_wf
∀[A:coSet{i:l}]. ∀[P:{a:coSet{i:l}| (a ∈ A)} ⟶ ℙ]. (∀a∈A.P[a] ∈ ℙ)
Proof
Definitions occuring in Statement :
allsetmem: ∀a∈A.P[a],
setmem: (x ∈ s),
coSet: coSet{i:l},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
allsetmem: ∀a∈A.P[a],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
set-item_wf,
set-item-mem,
setmem_wf,
coSet_wf,
set-dom_wf,
all_wf
Rules used in proof :
isect_memberEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_set_memberEquality,
dependent_functionElimination,
universeEquality,
cumulativity,
because_Cache,
setEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:coSet\{i:l\}]. \mforall{}[P:\{a:coSet\{i:l\}| (a \mmember{} A)\} {}\mrightarrow{} \mBbbP{}]. (\mforall{}a\mmember{}A.P[a] \mmember{} \mBbbP{})
Date html generated:
2018_07_29-AM-10_00_32
Last ObjectModification:
2018_07_18-PM-01_05_43
Theory : constructive!set!theory
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