Nuprl Lemma : equal-bag-size-le1

`∀[T:Type]. ∀[as,bs:bag(T)].`
`  uiff(as = bs ∈ bag(T);(as = {} ∈ bag(T) `⇐⇒` bs = {} ∈ bag(T))`
`  ∧ ((#(as) = 1 ∈ ℤ) `` (#(bs) = 1 ∈ ℤ) `` (only(as) = only(bs) ∈ T))) `
`  supposing (#(as) ≤ 1) ∧ (#(bs) ≤ 1)`

Proof

Definitions occuring in Statement :  bag-only: `only(bs)` bag-size: `#(bs)` empty-bag: `{}` bag: `bag(T)` uiff: `uiff(P;Q)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` iff: `P `⇐⇒` Q` implies: `P `` Q` and: `P ∧ Q` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` implies: `P `` Q` prop: `ℙ` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` nat: `ℕ` less_than: `a < b` squash: `↓T` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` true: `True` le: `A ≤ B` less_than': `less_than'(a;b)` guard: `{T}` cand: `A c∧ B`
Lemmas referenced :  equal-wf-T-base bag_wf and_wf equal_wf bag-only_wf bag-size_wf decidable__lt iff_wf le_wf decidable__le nat_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf empty-bag_wf bag-size-zero bag_size_empty_lemma less_than_wf squash_wf true_wf iff_weakening_equal decidable__equal_int intformeq_wf int_formula_prop_eq_lemma single-bag_wf bag-size-one
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin independent_pairFormation lambdaFormation equalityTransitivity equalitySymmetry hypothesis extract_by_obid isectElimination cumulativity hypothesisEquality baseClosed because_Cache dependent_set_memberEquality applyLambdaEquality setElimination rename independent_isectElimination hyp_replacement intEquality applyEquality sqequalRule independent_pairEquality lambdaEquality dependent_functionElimination axiomEquality natural_numberEquality unionElimination productEquality functionEquality isect_memberEquality imageElimination dependent_pairFormation int_eqEquality voidElimination voidEquality computeAll independent_functionElimination imageMemberEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[as,bs:bag(T)].
uiff(as  =  bs;(as  =  \{\}  \mLeftarrow{}{}\mRightarrow{}  bs  =  \{\})  \mwedge{}  ((\#(as)  =  1)  {}\mRightarrow{}  (\#(bs)  =  1)  {}\mRightarrow{}  (only(as)  =  only(bs))))
supposing  (\#(as)  \mleq{}  1)  \mwedge{}  (\#(bs)  \mleq{}  1)

Date html generated: 2017_10_01-AM-08_52_42
Last ObjectModification: 2017_07_26-PM-04_34_11

Theory : bags

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