Nuprl Lemma : subtype_rel_isect2
∀[A,B,C:Type]. uiff(A ⊆r B ⋂ C;(A ⊆r B) ∧ (A ⊆r C))
Proof
Definitions occuring in Statement :
isect2: T1 ⋂ T2,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
isect2: T1 ⋂ T2,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
ifthenelse: if b then t else f fi ,
bool: 𝔹,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
btrue: tt,
bfalse: ff
Lemmas referenced :
uall_wf,
bool_wf,
subtype_rel_wf,
ifthenelse_wf,
and_wf,
iff_weakening_uiff,
subtype_rel_isect,
uiff_wf,
btrue_wf,
bfalse_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
independent_pairFormation,
isect_memberFormation,
introduction,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
lemma_by_obid,
isectElimination,
lambdaEquality,
hypothesisEquality,
instantiate,
universeEquality,
unionElimination,
isect_memberEquality,
because_Cache,
addLevel,
independent_isectElimination,
isectEquality,
independent_functionElimination,
cumulativity,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A,B,C:Type]. uiff(A \msubseteq{}r B \mcap{} C;(A \msubseteq{}r B) \mwedge{} (A \msubseteq{}r C))
Date html generated:
2016_05_13-PM-03_58_04
Last ObjectModification:
2015_12_26-AM-10_51_33
Theory : bool_1
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