Nuprl Lemma : not-Sierpinski-top
∀[x:Sierpinski]. ((¬(x = ⊤ ∈ Sierpinski)) ⇒ (x = ⊥ ∈ Sierpinski))
Proof
Definitions occuring in Statement :
Sierpinski: Sierpinski,
Sierpinski-top: ⊤,
Sierpinski-bottom: ⊥,
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
Sierpinski: Sierpinski,
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
Sierpinski-bottom: ⊥,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
Sierpinski-unequal-1,
Sierpinski_wf,
quotient-member-eq,
nat_wf,
bool_wf,
iff_wf,
equal-wf-T-base,
two-class-equiv-rel,
Sierpinski-bottom_wf,
eqtt_to_assert,
Sierpinski-top_wf,
equal-wf-base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
not_wf,
assert_elim,
btrue_neq_bfalse
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
isect_memberFormation,
lambdaFormation,
pointwiseFunctionalityForEquality,
hypothesis,
sqequalRule,
pertypeElimination,
isectElimination,
functionEquality,
lambdaEquality,
hypothesisEquality,
baseClosed,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairFormation,
functionExtensionality,
rename,
applyEquality,
unionElimination,
equalityElimination,
voidElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
productEquality,
axiomEquality,
applyLambdaEquality
Latex:
\mforall{}[x:Sierpinski]. ((\mneg{}(x = \mtop{})) {}\mRightarrow{} (x = \mbot{}))
Date html generated:
2019_10_31-AM-06_35_29
Last ObjectModification:
2017_07_28-AM-09_11_56
Theory : synthetic!topology
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