### Nuprl Lemma : bag-member-subtype-2

`∀[A:Type]. ∀b:bag(A). ∀x:A.  (x ↓∈ b `` x ↓∈ b)`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag: `bag(T)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` bag-member: `x ↓∈ bs` squash: `↓T` sq_stable: `SqStable(P)` exists: `∃x:A. B[x]` and: `P ∧ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` l_all: `(∀x∈L.P[x])` cand: `A c∧ B` subtype_rel: `A ⊆r B` respects-equality: `respects-equality(S;T)` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` less_than: `a < b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bag: `bag(T)` quotient: `x,y:A//B[x; y]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` label: `...\$L... t` permutation: `permutation(T;L1;L2)`
Lemmas referenced :  bag-member_wf bag_wf istype-universe sq_stable__bag-member bag-subtype list-set-type2 int_seg_wf length_wf select_member subtype-respects-equality list_wf list-subtype-bag l_member_wf select_wf int_seg_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma l_member-settype respects-equality-bag respects-equality-set-trivial permutation_wf permutation_transitivity l_all_functionality_wrt_permutation permutation_inversion bag-member-select quotient-member-eq permutation-equiv inject_wf permute_list_wf strong-subtype-equal-lists strong-subtype-set3 strong-subtype-self
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut lambdaFormation_alt universeIsType extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis sqequalRule lambdaEquality_alt dependent_functionElimination imageElimination imageMemberEquality baseClosed functionIsTypeImplies inhabitedIsType instantiate universeEquality setEquality dependent_set_memberEquality_alt equalityTransitivity equalitySymmetry independent_functionElimination productElimination independent_isectElimination natural_numberEquality dependent_pairFormation_alt independent_pairFormation productIsType equalityIstype because_Cache setElimination rename unionElimination approximateComputation int_eqEquality isect_memberEquality_alt voidElimination setIsType pointwiseFunctionalityForEquality pertypeElimination promote_hyp sqequalBase applyEquality

Latex:
\mforall{}[A:Type].  \mforall{}b:bag(A).  \mforall{}x:A.    (x  \mdownarrow{}\mmember{}  b  {}\mRightarrow{}  x  \mdownarrow{}\mmember{}  b)

Date html generated: 2019_10_15-AM-11_02_10
Last ObjectModification: 2018_11_30-PM-00_12_51

Theory : bags

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