Nuprl Lemma : rleq_antisymmetry

[x,y:ℝ].  (x y) supposing ((y ≤ x) and (x ≤ y))


Definitions occuring in Statement :  rleq: x ≤ y req: y real: uimplies: supposing a uall: [x:A]. B[x]
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a iff: ⇐⇒ Q and: P ∧ Q implies:  Q req: y all: x:A. B[x] prop: real: bool: 𝔹 unit: Unit it: btrue: tt uiff: uiff(P;Q) less_than: a < b less_than': less_than'(a;b) top: Top true: True squash: T not: ¬A false: False nat_plus: + le: A ≤ B decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] bfalse: ff sq_type: SQType(T) guard: {T} bnot: ¬bb ifthenelse: if then else fi  assert: b
Lemmas referenced :  rleq-iff4 nat_plus_wf req_witness rleq_wf real_wf absval_unfold subtract_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermSubtract_wf itermVar_wf itermConstant_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_term_value_constant_lemma int_term_value_add_lemma int_formula_prop_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot itermMinus_wf int_term_value_minus_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis productElimination independent_functionElimination lambdaFormation dependent_functionElimination because_Cache sqequalRule isect_memberEquality equalityTransitivity equalitySymmetry applyEquality setElimination rename minusEquality natural_numberEquality unionElimination equalityElimination independent_isectElimination lessCases sqequalAxiom independent_pairFormation voidElimination voidEquality imageMemberEquality baseClosed imageElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality computeAll promote_hyp instantiate cumulativity

\mforall{}[x,y:\mBbbR{}].    (x  =  y)  supposing  ((y  \mleq{}  x)  and  (x  \mleq{}  y))

Date html generated: 2017_10_03-AM-08_25_03
Last ObjectModification: 2017_07_28-AM-07_23_40

Theory : reals

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