Nuprl Lemma : sub-set_wf2
∀[s:Set{i:l}]. ∀[P:{a:Set{i:l}| (a ∈ s)} ⟶ ℙ]. ({a ∈ s | P[a]} ∈ Set{i:l})
Proof
Definitions occuring in Statement :
sub-set: {a ∈ s | P[a]},
Set: Set{i:l},
setmem: (x ∈ s),
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 :
pi1: fst(t),
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
Wsup: Wsup(a;b),
mk-set: f"(T),
subtype_rel: A ⊆r B,
sub-set: {a ∈ s | P[a]},
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
set-subtype-coSet,
Set_wf,
setmem_wf,
setmem-mk-set,
mk-set_wf,
set-subtype,
subtype-set
Rules used in proof :
isect_memberEquality,
universeEquality,
cumulativity,
setEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
functionExtensionality,
lambdaEquality,
dependent_set_memberEquality,
dependent_functionElimination,
because_Cache,
productEquality,
isectElimination,
rename,
thin,
productElimination,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
hypothesis,
extract_by_obid,
hypothesis_subsumption,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[s:Set\{i:l\}]. \mforall{}[P:\{a:Set\{i:l\}| (a \mmember{} s)\} {}\mrightarrow{} \mBbbP{}]. (\{a \mmember{} s | P[a]\} \mmember{} Set\{i:l\})
Date html generated:
2018_07_29-AM-09_52_22
Last ObjectModification:
2018_07_18-AM-10_18_35
Theory : constructive!set!theory
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