Nuprl Lemma : equiv_rel_isect2
∀[A,B:Type]. ∀E:A ⟶ A ⟶ ℙ. (EquivRel(A;x,y.E[x;y]) ⇒ EquivRel(A ⋂ B;x,y.E[x;y]))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
isect2: T1 ⋂ T2,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
equiv_rel_subtype,
isect2_wf,
isect2_subtype_rel,
equiv_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}. (EquivRel(A;x,y.E[x;y]) {}\mRightarrow{} EquivRel(A \mcap{} B;x,y.E[x;y]))
Date html generated:
2016_05_13-PM-04_15_03
Last ObjectModification:
2015_12_26-AM-11_30_01
Theory : rel_1
Home
Index