Nuprl Lemma : set-predicate-iff
∀[s:Set{i:l}]. ∀[P:{x:Set{i:l}| (x ∈ s)} ⟶ ℙ'].
(set-predicate{i:l}(s;x.P[x]) ⇐⇒ ∀a1,a2:Set{i:l}. ((a1 ∈ s) ⇒ (a2 ∈ s) ⇒ seteq(a1;a2) ⇒ P[a1] ⇒ P[a2]))
Proof
Definitions occuring in Statement :
set-predicate: set-predicate{i:l}(s;a.P[a]),
Set: Set{i:l},
setmem: (x ∈ s),
seteq: seteq(s1;s2),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
guard: {T},
exists: ∃x:A. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
set-predicate: set-predicate{i:l}(s;a.P[a]),
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coSet_wf,
coSet-mem-Set-implies-Set,
set-subtype-coSet,
seteq_wf,
all_wf,
setmem_wf,
Set_wf,
set-predicate_wf2
Rules used in proof :
independent_functionElimination,
dependent_pairFormation,
independent_isectElimination,
dependent_functionElimination,
universeEquality,
dependent_set_memberEquality,
functionEquality,
instantiate,
because_Cache,
cumulativity,
hypothesis,
setEquality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[s:Set\{i:l\}]. \mforall{}[P:\{x:Set\{i:l\}| (x \mmember{} s)\} {}\mrightarrow{} \mBbbP{}'].
(set-predicate\{i:l\}(s;x.P[x])
\mLeftarrow{}{}\mRightarrow{} \mforall{}a1,a2:Set\{i:l\}. ((a1 \mmember{} s) {}\mRightarrow{} (a2 \mmember{} s) {}\mRightarrow{} seteq(a1;a2) {}\mRightarrow{} P[a1] {}\mRightarrow{} P[a2]))
Date html generated:
2018_07_29-AM-09_52_12
Last ObjectModification:
2018_07_18-AM-10_10_43
Theory : constructive!set!theory
Home
Index