Nuprl Lemma : subtype_rel_b-union_iff
∀[A,B,C:Type]. uiff((B ⋃ C) ⊆r A;(B ⊆r A) ∧ (C ⊆r A))
Proof
Definitions occuring in Statement :
b-union: A ⋃ B,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t)
Lemmas referenced :
subtype_rel_b-union-left,
subtype_rel_transitivity,
b-union_wf,
subtype_rel_b-union-right,
subtype_rel_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
independent_isectElimination,
sqequalRule,
productElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaEquality,
imageElimination,
unionElimination,
equalityElimination,
applyEquality
Latex:
\mforall{}[A,B,C:Type]. uiff((B \mcup{} C) \msubseteq{}r A;(B \msubseteq{}r A) \mwedge{} (C \msubseteq{}r A))
Date html generated:
2016_05_13-PM-03_57_54
Last ObjectModification:
2015_12_26-AM-10_52_04
Theory : bool_1
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