### Nuprl Lemma : id-graph-edge-implies-member

`∀S:Id List. ∀G:Graph(S). ∀i:{i:Id| (i ∈ S)} . ∀j:Id.  ((i⟶j)∈G `` (j ∈ S))`

Proof

Definitions occuring in Statement :  id-graph-edge: `(i⟶j)∈G` id-graph: `Graph(S)` Id: `Id` l_member: `(x ∈ l)` list: `T List` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} `
Definitions unfolded in proof :  id-graph-edge: `(i⟶j)∈G` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` id-graph: `Graph(S)` subtype_rel: `A ⊆r B` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` cand: `A c∧ B` Id: `Id` sq_type: `SQType(T)` guard: `{T}` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  equal_wf decidable__equal_Id sq_stable__l_member int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties select_wf atom2_subtype_base subtype_base_sq list_wf id-graph_wf set_wf subtype_rel_list Id_wf l_member_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesis hypothesisEquality applyEquality setEquality independent_isectElimination lambdaEquality setElimination rename because_Cache productElimination instantiate cumulativity dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination natural_numberEquality unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll introduction imageMemberEquality baseClosed imageElimination

Latex:
\mforall{}S:Id  List.  \mforall{}G:Graph(S).  \mforall{}i:\{i:Id|  (i  \mmember{}  S)\}  .  \mforall{}j:Id.    ((i{}\mrightarrow{}j)\mmember{}G  {}\mRightarrow{}  (j  \mmember{}  S))

Date html generated: 2016_05_14-PM-03_37_49
Last ObjectModification: 2016_01_14-PM-11_19_16

Theory : decidable!equality

Home Index