Nuprl Lemma : equipollent-decidable-equal
∀[A,B:Type]. (A ~ B ⇒ (∀x,y:B. Dec(x = y ∈ B)) ⇒ {∀x,y:A. Dec(x = y ∈ A)})
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
and: P ∧ Q,
biject: Bij(A;B;f),
inject: Inj(A;B;f)
Lemmas referenced :
all_wf,
decidable_wf,
equal_wf,
exists_wf,
biject_wf,
not_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesisEquality,
cut,
lemma_by_obid,
isectElimination,
lambdaEquality,
hypothesis,
functionEquality,
universeEquality,
dependent_functionElimination,
applyEquality,
unionElimination,
inlFormation,
inrFormation,
independent_functionElimination,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
rename,
setEquality,
voidElimination
Latex:
\mforall{}[A,B:Type]. (A \msim{} B {}\mRightarrow{} (\mforall{}x,y:B. Dec(x = y)) {}\mRightarrow{} \{\mforall{}x,y:A. Dec(x = y)\})
Date html generated:
2016_05_14-PM-03_59_45
Last ObjectModification:
2015_12_26-PM-07_44_44
Theory : equipollence!!cardinality!
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