### Nuprl Lemma : l_member-set2

`∀[T:Type]. ∀[P:T ⟶ ℙ].  ∀L:{x:T| P[x]}  List. ∀x:T.  ((x ∈ L) `` (x ∈ {x:T| P[x]} ))`

Proof

Definitions occuring in Statement :  l_member: `(x ∈ l)` list: `T List` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` so_apply: `x[s]` guard: `{T}` so_lambda: `λ2x.t[x]` or: `P ∨ Q` iff: `P `⇐⇒` Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf l_member_wf subtype_rel_list equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity set_wf list-cases nil_member nil_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int cons_member and_wf cons_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry cumulativity applyEquality setEquality functionExtensionality because_Cache unionElimination productElimination promote_hyp hypothesis_subsumption applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination hyp_replacement functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].    \mforall{}L:\{x:T|  P[x]\}    List.  \mforall{}x:T.    ((x  \mmember{}  L)  {}\mRightarrow{}  (x  \mmember{}  \{x:T|  P[x]\}  ))

Date html generated: 2017_04_17-AM-07_25_18
Last ObjectModification: 2017_02_27-PM-04_04_28

Theory : list_1

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