Nuprl Lemma : Sierpinski-equal-bottom
∀[x:ℕ ⟶ 𝔹]. uiff(x = ⊥ ∈ Sierpinski;x = ⊥ ∈ (ℕ ⟶ 𝔹))
Proof
Definitions occuring in Statement :
Sierpinski: Sierpinski,
Sierpinski-bottom: ⊥,
nat: ℕ,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
Sierpinski: Sierpinski,
quotient: x,y:A//B[x; y],
cand: A c∧ B,
iff: P ⇐⇒ Q,
squash: ↓T,
guard: {T},
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
equal_wf,
iff_weakening_equal,
member_wf,
nat_wf,
bool_wf,
equal-wf-base,
iff_wf,
equal-wf-T-base,
Sierpinski_wf,
subtype-Sierpinski,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
pertypeElimination,
productElimination,
thin,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
hypothesisEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productEquality,
functionEquality,
functionExtensionality,
independent_pairEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbB{}]. uiff(x = \mbot{};x = \mbot{})
Date html generated:
2019_10_31-AM-06_36_25
Last ObjectModification:
2017_07_28-AM-09_12_12
Theory : synthetic!topology
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