Nuprl Lemma : member-fset-union
∀[T:Type]. ∀eq:EqDecider(T). ∀x,y:fset(T). ∀a:T. (a ∈ x ⋃ y ⇐⇒ a ∈ x ∨ a ∈ y)
Proof
Definitions occuring in Statement :
fset-union: x ⋃ y,
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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
fset: fset(T),
prop: ℙ,
quotient: x,y:A//B[x; y],
not: ¬A,
fset-union: x ⋃ y,
fset-member: a ∈ s,
false: False,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
true: True,
rev_implies: P ⇐ Q
Lemmas referenced :
decidable__or,
fset-member_wf,
decidable__fset-member,
list_wf,
set-equal_wf,
set-equal-reflex,
assert-deq-member,
l-union_wf,
equal_wf,
subtype_base_sq,
int_subtype_base,
fset-union_wf,
fset-member_witness,
fset_wf,
deq_wf,
istype-universe,
member-union,
istype-assert,
deq-member_wf,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
unionElimination,
Error :universeIsType,
promote_hyp,
Error :inhabitedIsType,
pointwiseFunctionality,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
Error :productIsType,
Error :equalityIsType4,
intEquality,
natural_numberEquality,
instantiate,
cumulativity,
independent_isectElimination,
Error :unionIsType,
universeEquality,
Error :inlFormation_alt,
Error :inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}x,y:fset(T). \mforall{}a:T. (a \mmember{} x \mcup{} y \mLeftarrow{}{}\mRightarrow{} a \mmember{} x \mvee{} a \mmember{} y)
Date html generated:
2019_06_20-PM-01_58_42
Last ObjectModification:
2018_11_23-PM-02_42_33
Theory : finite!sets
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