### Nuprl Lemma : empty-bag-iff-size

`∀[T:Type]. ∀[bs:bag(T)].  uiff(bs = {} ∈ bag(T);#(bs) ≤ 0)`

Proof

Definitions occuring in Statement :  bag-size: `#(bs)` empty-bag: `{}` bag: `bag(T)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` le: `A ≤ B` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` le: `A ≤ B` not: `¬A` implies: `P `` Q` false: `False` prop: `ℙ` subtype_rel: `A ⊆r B` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` squash: `↓T` guard: `{T}` less_than: `a < b`
Lemmas referenced :  bag_wf less_than'_wf bag-size_wf nat_wf equal-wf-T-base le_wf iff_weakening_uiff all_wf not_wf bag-member_wf empty-bag-iff-no-member uiff_wf decidable__le decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_wf bag-member-iff-size
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality isect_memberEquality isectElimination hypothesisEquality lambdaEquality dependent_functionElimination because_Cache axiomEquality equalityTransitivity hypothesis equalitySymmetry extract_by_obid cumulativity voidElimination natural_numberEquality applyEquality setElimination rename baseClosed universeEquality addLevel independent_pairFormation independent_isectElimination independent_functionElimination instantiate lambdaFormation unionElimination dependent_pairFormation int_eqEquality intEquality voidEquality computeAll imageElimination imageMemberEquality

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].    uiff(bs  =  \{\};\#(bs)  \mleq{}  0)

Date html generated: 2017_10_01-AM-08_53_26
Last ObjectModification: 2017_07_26-PM-04_35_07

Theory : bags

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