Nuprl Lemma : bag-member-subtype
∀[A,B:Type]. ∀b:bag(A). ∀x:A. (x ↓∈ b ⇒ x ↓∈ b) supposing A ⊆r B
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
squash: ↓T,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
prop: ℙ,
bag-member: x ↓∈ bs,
and: P ∧ Q,
cand: A c∧ B
Lemmas referenced :
bag_to_squash_list,
sq_stable__bag-member,
subtype_rel_bag,
bag-member_wf,
list-subtype-bag,
subtype_rel_list,
l_member_subtype,
equal_wf,
bag_wf,
l_member_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
imageElimination,
applyEquality,
hypothesis,
sqequalRule,
independent_isectElimination,
independent_functionElimination,
productElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
cumulativity,
rename,
dependent_pairFormation,
independent_pairFormation,
dependent_functionElimination,
productEquality,
lambdaEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality,
equalityTransitivity,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}b:bag(A). \mforall{}x:A. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} x \mdownarrow{}\mmember{} b) supposing A \msubseteq{}r B
Date html generated:
2017_10_01-AM-08_53_12
Last ObjectModification:
2017_07_26-PM-04_34_47
Theory : bags
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