Nuprl Lemma : natset-setmem-natset
∀n,m:ℕ. ((natset(n) ∈ natset(m)) ⇐⇒ n < m)
Proof
Definitions occuring in Statement :
natset: natset(n),
setmem: (x ∈ s),
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
rev_implies: P ⇐ Q,
so_apply: x[s],
lelt: i ≤ j < k,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
setmem-irreflexive,
setmem_functionality,
seteq_weakening,
lelt_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
decidable__lt,
iff_wf,
all_wf,
setmem_wf,
setmem-natset,
nat_wf,
less_than_wf,
le_wf,
natset_wf,
seteq_wf,
int_seg_wf,
exists_wf
Rules used in proof :
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
unionElimination,
independent_functionElimination,
dependent_functionElimination,
impliesFunctionality,
allFunctionality,
addLevel,
because_Cache,
dependent_set_memberEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
isectElimination,
extract_by_obid,
introduction,
hypothesis,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}n,m:\mBbbN{}. ((natset(n) \mmember{} natset(m)) \mLeftarrow{}{}\mRightarrow{} n < m)
Date html generated:
2018_05_29-PM-01_49_42
Last ObjectModification:
2018_05_24-PM-11_56_57
Theory : constructive!set!theory
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