Nuprl Lemma : sp-join-is-bottom
∀[x,y:Sierpinski]. (x ∨ y = ⊥ ∈ Sierpinski ⇐⇒ (x = ⊥ ∈ Sierpinski) ∧ (y = ⊥ ∈ Sierpinski))
Proof
Definitions occuring in Statement :
sp-join: f ∨ g,
Sierpinski: Sierpinski,
Sierpinski-bottom: ⊥,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: 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],
prop: ℙ,
rev_implies: P ⇐ Q,
sp-join: f ∨ g,
cand: A c∧ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
not: ¬A,
false: False,
or: P ∨ Q,
guard: {T},
true: True,
Sierpinski-bottom: ⊥,
bfalse: ff,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
squash: ↓T,
subtype_rel: A ⊆r B
Lemmas referenced :
Sierpinski-unequal-1,
Sierpinski_wf,
equal-wf-base,
iff_wf,
equal-wf-T-base,
nat_wf,
bool_wf,
sp-join_wf,
quotient-member-eq,
two-class-equiv-rel,
member_wf,
bor_wf,
equal-Sierpinski-bottom,
assert_of_bor,
or_wf,
assert_wf,
Sierpinski-bottom_wf,
bfalse_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
pointwiseFunctionalityForEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productEquality,
isectElimination,
because_Cache,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
functionEquality,
equalityElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
independent_isectElimination,
independent_functionElimination,
applyEquality,
allFunctionality,
promote_hyp,
voidElimination,
inlFormation,
inrFormation,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[x,y:Sierpinski]. (x \mvee{} y = \mbot{} \mLeftarrow{}{}\mRightarrow{} (x = \mbot{}) \mwedge{} (y = \mbot{}))
Date html generated:
2019_10_31-AM-06_35_55
Last ObjectModification:
2017_07_28-AM-09_12_01
Theory : synthetic!topology
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