Nuprl Lemma : f-union-subset
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[x,y,z:fset(A)]. uiff(x ⋃ y ⊆ z;x ⊆ z ∧ y ⊆ z)
Proof
Definitions occuring in Statement :
fset-union: x ⋃ y,
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
f-subset: xs ⊆ ys,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
or: P ∨ Q,
prop: ℙ,
guard: {T}
Lemmas referenced :
member-fset-union,
fset-member_wf,
fset-member_witness,
f-subset_wf,
fset-union_wf,
and_wf,
fset_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
lambdaFormation,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lemma_by_obid,
isectElimination,
because_Cache,
productElimination,
independent_functionElimination,
inlFormation,
sqequalRule,
inrFormation,
independent_pairEquality,
lambdaEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
unionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[x,y,z:fset(A)]. uiff(x \mcup{} y \msubseteq{} z;x \msubseteq{} z \mwedge{} y \msubseteq{} z)
Date html generated:
2016_05_14-PM-03_38_43
Last ObjectModification:
2015_12_26-PM-06_42_04
Theory : finite!sets
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