Nuprl Lemma : equipollent_transitivity
∀[A,B,C:Type]. (A ~ B ⇒ B ~ C ⇒ A ~ C)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
biject: Bij(A;B;f),
and: P ∧ Q,
cand: A c∧ B,
surject: Surj(A;B;f),
inject: Inj(A;B;f),
all: ∀x:A. B[x],
guard: {T},
compose: f o g,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
exists_wf,
biject_wf,
compose_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
cumulativity,
hypothesisEquality,
lambdaEquality,
functionExtensionality,
applyEquality,
hypothesis,
universeEquality,
rename,
dependent_pairFormation,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[A,B,C:Type]. (A \msim{} B {}\mRightarrow{} B \msim{} C {}\mRightarrow{} A \msim{} C)
Date html generated:
2017_04_17-AM-09_30_53
Last ObjectModification:
2017_02_27-PM-05_31_06
Theory : equipollence!!cardinality!
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