Nuprl Lemma : finite'_functionality_wrt_equipollent
∀[A,B:Type]. (A ~ B ⇒ (finite'(A) ⇐⇒ finite'(B)))
Proof
Definitions occuring in Statement :
finite': finite'(T),
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
finite': finite'(T),
equipollent: A ~ B,
exists: ∃x:A. B[x],
biject: Bij(A;B;f),
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
surject: Surj(A;B;f),
compose: f o g,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
inject: Inj(A;B;f),
guard: {T}
Lemmas referenced :
inject_wf,
finite'_wf,
equipollent_wf,
compose_wf,
equal_wf,
exists_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
functionEquality,
universeEquality,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
dependent_pairFormation,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
equalityTransitivity,
because_Cache,
lambdaEquality
Latex:
\mforall{}[A,B:Type]. (A \msim{} B {}\mRightarrow{} (finite'(A) \mLeftarrow{}{}\mRightarrow{} finite'(B)))
Date html generated:
2016_10_21-AM-11_00_00
Last ObjectModification:
2016_08_06-PM-02_33_11
Theory : equipollence!!cardinality!
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