Nuprl Lemma : cardinality-le_functionality_wrt_equipollence
∀[A,B:Type]. ∀[k:ℕ]. (A ~ B ⇒ {|A| ≤ k ⇐⇒ |B| ≤ k})
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
cardinality-le: |T| ≤ n,
nat: ℕ,
uall: ∀[x:A]. B[x],
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equipollent: A ~ B,
exists: ∃x:A. B[x],
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
cardinality-le: |T| ≤ n,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
nat: ℕ,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
all: ∀x:A. B[x],
compose: f o g,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
cardinality-le_wf,
equipollent_wf,
nat_wf,
compose_wf,
int_seg_wf,
surject_wf,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
biject-inverse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
universeEquality,
dependent_pairFormation,
natural_numberEquality,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[k:\mBbbN{}]. (A \msim{} B {}\mRightarrow{} \{|A| \mleq{} k \mLeftarrow{}{}\mRightarrow{} |B| \mleq{} k\})
Date html generated:
2018_05_21-PM-00_52_27
Last ObjectModification:
2018_05_19-AM-06_39_38
Theory : equipollence!!cardinality!
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