Nuprl Lemma : bag-in-subtype2
∀[A,B:Type]. ∀[b:bag(B)]. b ∈ bag(A) supposing ∀x:B. (x ↓∈ b ⇒ (∃a:bag(A). x ↓∈ a)) supposing strong-subtype(A;B)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B
Lemmas referenced :
bag-in-subtype,
bag-member_wf,
bag_wf,
subtype_rel_bag,
strong-subtype_wf,
istype-universe,
bag-member-strong-subtype
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
axiomEquality,
functionIsType,
productIsType,
applyEquality,
productElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality
Latex:
\mforall{}[A,B:Type].
\mforall{}[b:bag(B)]. b \mmember{} bag(A) supposing \mforall{}x:B. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (\mexists{}a:bag(A). x \mdownarrow{}\mmember{} a))
supposing strong-subtype(A;B)
Date html generated:
2019_10_15-AM-11_01_29
Last ObjectModification:
2019_08_15-PM-03_50_04
Theory : bags
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