Nuprl Lemma : equipollent-subtype
∀[T,A:Type].
(∀[a,b:ℕ]. (a ≤ b) supposing (T ~ ℕb and A ~ ℕa)) supposing
((∀a1,a2:A. ((a1 = a2 ∈ T) ⇒ (a1 = a2 ∈ A))) and
(A ⊆r T))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
equipollent: A ~ B,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
inject: Inj(A;B;f),
compose: f o g,
guard: {T},
biject: Bij(A;B;f)
Lemmas referenced :
less_than'_wf,
equipollent_wf,
int_seg_wf,
nat_wf,
all_wf,
equal_wf,
subtype_rel_wf,
equipollent_inversion,
compose_wf,
subtype_rel_dep_function,
subtype_rel_self,
inject_wf,
pigeon-hole
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
lemma_by_obid,
isectElimination,
setElimination,
rename,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
isect_memberEquality,
functionEquality,
applyEquality,
universeEquality,
voidElimination,
independent_functionElimination,
dependent_pairFormation,
independent_isectElimination,
lambdaFormation
Latex:
\mforall{}[T,A:Type].
(\mforall{}[a,b:\mBbbN{}]. (a \mleq{} b) supposing (T \msim{} \mBbbN{}b and A \msim{} \mBbbN{}a)) supposing
((\mforall{}a1,a2:A. ((a1 = a2) {}\mRightarrow{} (a1 = a2))) and
(A \msubseteq{}r T))
Date html generated:
2016_05_14-PM-04_03_55
Last ObjectModification:
2015_12_26-PM-07_42_10
Theory : equipollence!!cardinality!
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