Nuprl Lemma : set-equal-l_contains
∀[T:Type]. ∀x,y:T List. (set-equal(T;x;y) ⇐⇒ x ⊆ y ∧ y ⊆ x)
Proof
Definitions occuring in Statement :
set-equal: set-equal(T;x;y),
l_contains: A ⊆ B,
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
l_contains: A ⊆ B,
set-equal: set-equal(T;x;y),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
l_member_wf,
all_wf,
iff_wf,
l_all_iff,
l_all_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
lambdaEquality,
productElimination,
productEquality,
functionEquality,
addLevel,
independent_functionElimination,
dependent_functionElimination,
setElimination,
rename,
setEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}x,y:T List. (set-equal(T;x;y) \mLeftarrow{}{}\mRightarrow{} x \msubseteq{} y \mwedge{} y \msubseteq{} x)
Date html generated:
2019_06_20-PM-01_30_21
Last ObjectModification:
2018_08_24-PM-11_35_14
Theory : list_1
Home
Index