Nuprl Lemma : fset-member_witness
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a:T]. ∀[s:fset(T)]. (a ∈ s ⇒ (Ax ∈ a ∈ s))
Proof
Definitions occuring in Statement :
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
universe: Type,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
fset-member: a ∈ s,
fset: fset(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
set-equal: set-equal(T;x;y),
all: ∀x:A. B[x],
guard: {T}
Lemmas referenced :
fset-member_wf,
fset_wf,
deq_wf,
assert_witness,
bool_wf,
iff_imp_equal_bool,
deq-member_wf,
l_member_wf,
assert-deq-member,
assert_wf,
iff_wf,
equal-wf-base,
list_wf,
set-equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
universeEquality,
independent_functionElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
addLevel,
impliesFunctionality,
productEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[a:T]. \mforall{}[s:fset(T)]. (a \mmember{} s {}\mRightarrow{} (Ax \mmember{} a \mmember{} s))
Date html generated:
2016_05_14-PM-03_38_07
Last ObjectModification:
2015_12_26-PM-06_42_24
Theory : finite!sets
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