Nuprl Lemma : sub-bag-equal
∀[T:Type]. ∀[b1,b2:bag(T)]. (b1 = b2 ∈ bag(T)) supposing (sub-bag(T;b1;b2) and sub-bag(T;b2;b1))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
top: Top,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
bag_wf,
bag-append_wf,
bag-append-assoc,
bag-append-empty,
bag-subtype-list,
bag-append-cancel,
empty-bag_wf,
bag-append-eq-empty,
squash_wf,
true_wf,
iff_weakening_equal,
sub-bag_wf
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
cut,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
equalityTransitivity,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
dependent_functionElimination,
independent_isectElimination,
lambdaEquality,
imageElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
isect_memberFormation,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[b1,b2:bag(T)]. (b1 = b2) supposing (sub-bag(T;b1;b2) and sub-bag(T;b2;b1))
Date html generated:
2017_10_01-AM-08_53_08
Last ObjectModification:
2017_07_26-PM-04_34_40
Theory : bags
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