Nuprl Lemma : subtype_rel_b-union
∀[A1,B1,A2,B2:Type]. (A1 ⋃ B1) ⊆r (A2 ⋃ B2) supposing (A1 ⊆r A2) ∧ (B1 ⊆r B2)
Proof
Definitions occuring in Statement :
b-union: A ⋃ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
b-union: A ⋃ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
subtype_rel_tunion,
bool_wf,
ifthenelse_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
instantiate,
hypothesisEquality,
universeEquality,
cumulativity,
because_Cache,
independent_isectElimination,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
productEquality,
isect_memberEquality
Latex:
\mforall{}[A1,B1,A2,B2:Type]. (A1 \mcup{} B1) \msubseteq{}r (A2 \mcup{} B2) supposing (A1 \msubseteq{}r A2) \mwedge{} (B1 \msubseteq{}r B2)
Date html generated:
2017_04_14-AM-07_30_21
Last ObjectModification:
2017_02_27-PM-02_59_05
Theory : bool_1
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