Nuprl Lemma : fset-pair-is-union
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x,y:T]. ({x,y} = {x} ⋃ {y} ∈ fset(T))
Proof
Definitions occuring in Statement :
fset-pair: {a,b},
fset-singleton: {x},
fset-union: x ⋃ y,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
fset-extensionality,
fset-pair_wf,
fset-union_wf,
fset-singleton_wf,
fset-member_witness,
member-fset-pair,
member-fset-singleton,
fset-member_wf,
or_wf,
equal_wf,
member-fset-union,
uiff_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
cumulativity,
hypothesis,
productElimination,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
rename,
dependent_functionElimination,
addLevel,
orFunctionality,
promote_hyp,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:T]. (\{x,y\} = \{x\} \mcup{} \{y\})
Date html generated:
2017_04_17-AM-09_19_00
Last ObjectModification:
2017_02_27-PM-05_22_25
Theory : finite!sets
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