### Nuprl Lemma : bag-member-single

`∀[T:Type]. ∀[x,y:T].  uiff(x ↓∈ {y};x = y ∈ T)`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` single-bag: `{x}` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` 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` prop: `ℙ` all: `∀x:A. B[x]` bag-member: `x ↓∈ bs` squash: `↓T` exists: `∃x:A. B[x]` single-bag: `{x}` bag: `bag(T)` quotient: `x,y:A//B[x; y]` cand: `A c∧ B` permutation: `permutation(T;L1;L2)` top: `Top` sq_type: `SQType(T)` implies: `P `` Q` guard: `{T}` l_member: `(x ∈ l)` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` less_than: `a < b` true: `True` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` select: `L[n]` cons: `[a / b]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k`
Lemmas referenced :  bag-member_wf single-bag_wf equal_wf member_wf list_wf cons_wf nil_wf permutation_wf length_wf length_of_cons_lemma length_of_nil_lemma permute_list_length subtype_base_sq int_subtype_base squash_wf true_wf select_wf le_wf less_than_wf false_wf length-singleton nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf intformeq_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_eq_lemma int_formula_prop_wf iff_weakening_equal permute_list_select subtype_rel_dep_function int_seg_wf int_seg_subtype decidable__le intformle_wf int_formula_prop_le_lemma int_seg_properties lelt_wf set_subtype_base non_neg_length length_wf_nat decidable__equal_int cons_member l_member_wf bag_wf list-subtype-bag
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality dependent_functionElimination sqequalRule imageElimination imageMemberEquality baseClosed productElimination independent_pairEquality isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry universeEquality pertypeElimination productEquality applyLambdaEquality voidElimination voidEquality instantiate intEquality independent_isectElimination independent_functionElimination applyEquality lambdaEquality natural_numberEquality lambdaFormation setElimination rename unionElimination dependent_pairFormation int_eqEquality computeAll dependent_set_memberEquality functionExtensionality hyp_replacement inlFormation

Latex:
\mforall{}[T:Type].  \mforall{}[x,y:T].    uiff(x  \mdownarrow{}\mmember{}  \{y\};x  =  y)

Date html generated: 2017_10_01-AM-08_53_32
Last ObjectModification: 2017_07_26-PM-04_35_11

Theory : bags

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