Nuprl Lemma : member-fset-singleton
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x,y:T]. uiff(y ∈ {x};y = x ∈ T)
Proof
Definitions occuring in Statement :
fset-singleton: {x},
fset-member: a ∈ s,
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
fset-singleton: {x},
fset-member: a ∈ s,
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
or: P ∨ Q,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
rev_implies: P ⇐ Q,
eqof: eqof(d)
Lemmas referenced :
deq_member_cons_lemma,
deq_member_nil_lemma,
false_wf,
equal_wf,
assert_wf,
bor_wf,
eqof_wf,
bfalse_wf,
or_wf,
uiff_wf,
fset-member_wf,
fset-singleton_wf,
deq_wf,
fset-member_witness,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
safe-assert-deq,
assert_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
independent_pairFormation,
isect_memberFormation,
unionElimination,
equalitySymmetry,
because_Cache,
axiomEquality,
rename,
inlFormation,
isectElimination,
cumulativity,
hypothesisEquality,
applyEquality,
universeEquality,
productElimination,
independent_pairEquality,
equalityTransitivity,
independent_functionElimination,
addLevel,
independent_isectElimination,
lambdaFormation,
orFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:T]. uiff(y \mmember{} \{x\};y = x)
Date html generated:
2017_04_17-AM-09_18_55
Last ObjectModification:
2017_02_27-PM-05_22_22
Theory : finite!sets
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