Nuprl Lemma : equipollent-subtract-one
∀a:ℕ. ∀i:ℕa. {x:ℕa| ¬(x = i ∈ ℕa)} ~ ℕa - 1
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
not: ¬A,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
equipollent-subtract,
false_wf,
le_wf,
equal_wf,
int_seg_wf,
equipollent-one,
nat_wf
Rules used in proof :
independent_functionElimination,
rename,
setElimination,
lambdaEquality,
isectElimination,
hypothesis,
independent_pairFormation,
sqequalRule,
natural_numberEquality,
dependent_set_memberEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:\mBbbN{}. \mforall{}i:\mBbbN{}a. \{x:\mBbbN{}a| \mneg{}(x = i)\} \msim{} \mBbbN{}a - 1
Date html generated:
2018_05_21-PM-00_53_32
Last ObjectModification:
2017_12_07-PM-06_24_44
Theory : equipollence!!cardinality!
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