Nuprl Lemma : equipollent-nsub
∀n,m:ℕ. (n = m ∈ ℤ ⇐⇒ ℕn ~ ℕm)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
biject: Bij(A;B;f),
exists: ∃x:A. B[x],
equipollent: A ~ B,
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j
Lemmas referenced :
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
pigeon-hole,
equipollent_inversion,
ext-eq_weakening,
equipollent_weakening_ext-eq,
subtype_base_sq,
int_subtype_base,
equal_wf,
equipollent_wf,
int_seg_wf,
nat_wf
Rules used in proof :
natural_numberEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
intEquality,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
independent_isectElimination,
cumulativity,
instantiate,
productElimination,
computeAll,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
unionElimination,
because_Cache
Latex:
\mforall{}n,m:\mBbbN{}. (n = m \mLeftarrow{}{}\mRightarrow{} \mBbbN{}n \msim{} \mBbbN{}m)
Date html generated:
2018_05_21-PM-00_52_32
Last ObjectModification:
2017_12_07-PM-06_17_35
Theory : equipollence!!cardinality!
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