Nuprl Lemma : not-Sierpinski-bottom
∀[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],
not: ¬A,
prop: ℙ,
rev_implies: P ⇐ Q,
false: False,
subtype_rel: A ⊆r B
Lemmas referenced :
subtype-Sierpinski,
not_wf,
equal-wf-T-base,
equal-wf-base,
Sierpinski-top_wf,
two-class-equiv-rel,
Sierpinski-bottom_wf,
equal_wf,
iff_wf,
bool_wf,
nat_wf,
quotient-member-eq,
Sierpinski_wf,
Sierpinski-unequal-1
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
isect_memberFormation,
introduction,
lambdaFormation,
pointwiseFunctionalityForEquality,
hypothesis,
sqequalRule,
pertypeElimination,
isectElimination,
functionEquality,
lambdaEquality,
hypothesisEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairFormation,
functionExtensionality,
voidElimination,
productEquality,
because_Cache,
baseClosed,
applyEquality,
axiomEquality
Latex:
\mforall{}[x:Sierpinski]. ((\mneg{}(x = \mbot{})) {}\mRightarrow{} (x = \mtop{}))
Date html generated:
2019_10_31-AM-06_35_31
Last ObjectModification:
2016_01_17-AM-09_36_00
Theory : synthetic!topology
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