Nuprl Lemma : one_one_corr_equipollent
∀[A,B:Type]. (1-1-Corresp(A;B) ⇐⇒ A ~ B)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
one_one_corr: 1-1-Corresp(A;B),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
one_one_corr: 1-1-Corresp(A;B),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
one_one_corr_wf,
sq_exists_wf,
inv_funs_wf,
pi1_wf,
pi2_wf,
equipollent-iff-inverse-funs,
equipollent_wf,
iff_wf,
exists_wf,
sq_stable__inv_funs
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productEquality,
functionEquality,
sqequalRule,
lambdaEquality,
addLevel,
independent_functionElimination,
because_Cache,
universeEquality,
dependent_set_memberFormation,
independent_pairEquality,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[A,B:Type]. (1-1-Corresp(A;B) \mLeftarrow{}{}\mRightarrow{} A \msim{} B)
Date html generated:
2019_06_20-PM-02_16_46
Last ObjectModification:
2018_08_24-PM-11_37_01
Theory : equipollence!!cardinality!
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