Nuprl Lemma : sp-join_wf
∀[f,g:Sierpinski]. (f ∨ g ∈ Sierpinski)
Proof
Definitions occuring in Statement :
sp-join: f ∨ g,
Sierpinski: Sierpinski,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Sierpinski: Sierpinski,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
sp-join: f ∨ g,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
not: ¬A,
false: False,
or: P ∨ Q,
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
bor: p ∨bq,
Sierpinski-bottom: ⊥,
bfalse: ff
Lemmas referenced :
Sierpinski_wf,
quotient-member-eq,
nat_wf,
bool_wf,
iff_wf,
equal-wf-T-base,
two-class-equiv-rel,
Sierpinski-bottom_wf,
bor_wf,
equal-wf-base,
equal-Sierpinski-bottom,
assert_wf,
assert_of_bor,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
thin,
isectElimination,
functionEquality,
lambdaEquality,
because_Cache,
hypothesisEquality,
baseClosed,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
independent_functionElimination,
independent_pairFormation,
lambdaFormation,
productEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isect_memberEquality,
hyp_replacement,
applyLambdaEquality,
functionExtensionality,
voidElimination,
allFunctionality,
promote_hyp,
inlFormation,
inrFormation,
rename
Latex:
\mforall{}[f,g:Sierpinski]. (f \mvee{} g \mmember{} Sierpinski)
Date html generated:
2019_10_31-AM-06_35_48
Last ObjectModification:
2017_07_28-AM-09_11_59
Theory : synthetic!topology
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