Nuprl Lemma : equipollent_functionality_wrt_ext-eq-left
∀[A1,A2,B1,B2:Type]. (A1 ~ B1 ⇐⇒ A2 ~ B2) supposing ((B1 = B2 ∈ Type) and A1 ≡ A2)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
guard: {T},
rev_implies: P ⇐ Q,
prop: ℙ
Lemmas referenced :
equipollent_functionality_wrt_ext-eq,
ext-eq_weakening,
equipollent_wf,
equal_wf,
ext-eq_wf,
equipollent_transitivity,
equipollent_weakening_ext-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
hypothesis,
rename,
independent_pairFormation,
lambdaFormation,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
independent_functionElimination,
instantiate,
universeEquality,
because_Cache,
equalitySymmetry
Latex:
\mforall{}[A1,A2,B1,B2:Type]. (A1 \msim{} B1 \mLeftarrow{}{}\mRightarrow{} A2 \msim{} B2) supposing ((B1 = B2) and A1 \mequiv{} A2)
Date html generated:
2016_05_14-PM-04_00_18
Last ObjectModification:
2015_12_26-PM-07_44_29
Theory : equipollence!!cardinality!
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