Nuprl Lemma : equipollent-singletons
∀[A,B:Type]. (singleton-type(A) ⇒ singleton-type(B) ⇒ A ~ B)
Proof
Definitions occuring in Statement :
singleton-type: singleton-type(A),
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
singleton-type: singleton-type(A),
exists: ∃x:A. B[x],
equipollent: A ~ B,
member: t ∈ T,
prop: ℙ,
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
surject: Surj(A;B;f),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
biject_wf,
singleton-type_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
trivial-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
dependent_pairFormation,
lambdaEquality,
hypothesisEquality,
cumulativity,
cut,
introduction,
extract_by_obid,
isectElimination,
functionExtensionality,
applyEquality,
hypothesis,
universeEquality,
independent_pairFormation,
sqequalRule,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
because_Cache
Latex:
\mforall{}[A,B:Type]. (singleton-type(A) {}\mRightarrow{} singleton-type(B) {}\mRightarrow{} A \msim{} B)
Date html generated:
2017_04_17-AM-09_31_36
Last ObjectModification:
2017_02_27-PM-05_31_36
Theory : equipollence!!cardinality!
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