Nuprl Lemma : sp-meet-is-top
∀[x,y:Sierpinski]. (x ∧ y = ⊤ ∈ Sierpinski ⇐⇒ (x = ⊤ ∈ Sierpinski) ∧ (y = ⊤ ∈ Sierpinski))
Proof
Definitions occuring in Statement :
sp-meet: f ∧ g,
Sierpinski: Sierpinski,
Sierpinski-top: ⊤,
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-meet: 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],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
Sierpinski-bottom: ⊥,
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
false: False,
Sierpinski-top: ⊤,
btrue: tt,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
assert: ↑b,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
coded-pair: coded-pair(m),
code-pair: code-pair(a;b),
triangular-num: t(n),
tsqrt: tsqrt(n),
isqrt: isqrt(x),
integer-sqrt-ext,
genrec-ap: genrec-ap,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
subtract: n - m
Lemmas referenced :
Sierpinski-unequal-1,
Sierpinski_wf,
equal-wf-base,
iff_wf,
equal-wf-T-base,
nat_wf,
bool_wf,
sp-meet_wf,
quotient-member-eq,
two-class-equiv-rel,
Sierpinski-bottom_wf,
equal_wf,
coded-pair_wf,
band_wf,
squash_wf,
true_wf,
iff_weakening_equal,
spread_to_pi12,
top_wf,
subtype_rel_product,
bfalse_wf,
member_wf,
band_ff_simp,
btrue_wf,
equal-Sierpinski-bottom,
code-pair_wf,
false_wf,
le_wf,
assert_of_bnot,
integer-sqrt-ext
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,
imageElimination,
equalityUniverse,
levelHypothesis,
spreadEquality,
functionExtensionality,
natural_numberEquality,
imageMemberEquality,
voidElimination,
voidEquality,
universeEquality,
dependent_set_memberEquality
Latex:
\mforall{}[x,y:Sierpinski]. (x \mwedge{} y = \mtop{} \mLeftarrow{}{}\mRightarrow{} (x = \mtop{}) \mwedge{} (y = \mtop{}))
Date html generated:
2019_10_31-AM-06_35_51
Last ObjectModification:
2017_07_28-AM-09_12_00
Theory : synthetic!topology
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