Nuprl Lemma : equipollent-cardinality-le
∀[A:Type]. ∀[k:ℕ]. (A ~ ℕk ⇒ |A| ≤ k)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
cardinality-le: |T| ≤ n,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
member: t ∈ T,
nat: ℕ,
and: P ∧ Q,
cardinality-le: |T| ≤ n,
prop: ℙ,
surject: Surj(A;B;f),
all: ∀x:A. B[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
biject-inverse,
int_seg_wf,
equipollent_wf,
nat_wf,
surject_wf,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
universeEquality,
dependent_pairFormation,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination
Latex:
\mforall{}[A:Type]. \mforall{}[k:\mBbbN{}]. (A \msim{} \mBbbN{}k {}\mRightarrow{} |A| \mleq{} k)
Date html generated:
2018_05_21-PM-00_52_24
Last ObjectModification:
2018_05_19-AM-06_39_31
Theory : equipollence!!cardinality!
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