Nuprl Lemma : member-fset-pair
∀[T:Type]. ∀eq:EqDecider(T). ∀x,y,z:T. uiff(z ∈ {x,y};(z = x ∈ T) ∨ (z = y ∈ T))
Proof
Definitions occuring in Statement :
fset-pair: {a,b},
fset-member: a ∈ s,
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
fset-pair: {a,b},
fset-member: a ∈ s,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
iff: P ⇐⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
rev_implies: P ⇐ Q,
eqof: eqof(d),
decidable: Dec(P),
not: ¬A,
false: False
Lemmas referenced :
deq-implies,
deq_member_cons_lemma,
deq_member_nil_lemma,
deq_wf,
or_wf,
equal_wf,
bor_wf,
eqof_wf,
bfalse_wf,
assert_wf,
false_wf,
uiff_wf,
assert_witness,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
safe-assert-deq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_functionElimination,
hypothesis,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesisEquality,
cumulativity,
universeEquality,
independent_pairFormation,
applyEquality,
addLevel,
productElimination,
rename,
independent_isectElimination,
orFunctionality,
unionElimination,
inlFormation,
inrFormation,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}x,y,z:T. uiff(z \mmember{} \{x,y\};(z = x) \mvee{} (z = y))
Date html generated:
2017_04_17-AM-09_18_59
Last ObjectModification:
2017_02_27-PM-05_22_29
Theory : finite!sets
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