Nuprl Lemma : equipollent-unit
∀[T:Type]. (T ⇒ T ~ Unit supposing ∀x,y:T. (x = y ∈ T))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
equipollent: A ~ B,
exists: ∃x:A. B[x],
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
surject: Surj(A;B;f),
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Lemmas referenced :
it_wf,
unit_wf2,
equal-unit,
unit_subtype_base,
biject_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
Error :lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
rename,
Error :dependent_pairFormation_alt,
closedConclusion,
extract_by_obid,
independent_pairFormation,
Error :equalityIsType4,
Error :universeIsType,
baseClosed,
isectElimination,
baseApply,
applyEquality,
Error :functionIsType,
Error :equalityIsType1,
universeEquality
Latex:
\mforall{}[T:Type]. (T {}\mRightarrow{} T \msim{} Unit supposing \mforall{}x,y:T. (x = y))
Date html generated:
2019_06_20-PM-02_17_01
Last ObjectModification:
2018_10_12-PM-06_04_39
Theory : equipollence!!cardinality!
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