Nuprl Lemma : member-fset-add
∀[T:Type]. ∀eq:EqDecider(T). ∀s:fset(T). ∀x,y:T. (x ∈ fset-add(eq;y;s) ⇐⇒ (x = y ∈ T) ∨ x ∈ s)
Proof
Definitions occuring in Statement :
fset-add: fset-add(eq;x;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
fset-add: fset-add(eq;x;s),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
or: P ∨ Q
Lemmas referenced :
or_wf,
equal_wf,
fset-member_wf,
member-fset-singleton,
fset-singleton_wf,
iff_wf,
member-fset-union,
fset-union_wf,
fset_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
because_Cache,
addLevel,
productElimination,
independent_functionElimination,
orFunctionality,
independent_isectElimination,
impliesFunctionality,
dependent_functionElimination,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}s:fset(T). \mforall{}x,y:T. (x \mmember{} fset-add(eq;y;s) \mLeftarrow{}{}\mRightarrow{} (x = y) \mvee{} x \mmember{} s)
Date html generated:
2017_04_17-AM-09_19_45
Last ObjectModification:
2017_02_27-PM-05_22_53
Theory : finite!sets
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