matita/matita/contribs/lambdadelta/ground/lib/subset_equivalence.ma

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  14
  15 include "ground/notation/relations/white_harrow_2.ma".
  16 include "ground/lib/subset_inclusion.ma".
  17
  18 (* EQUIVALENCE FOR SUBSETS **************************************************)
  19
  20 definition subset_eq (A): relation2 𝒫❨A❩ 𝒫❨A❩ ≝
  21            λu1,u2. ∧∧ u1 ⊆ u2 & u2 ⊆ u1.
  22
  23 interpretation
  24   "equivalence (subset)"
  25   'WhiteHArrow u1 u2 = (subset_eq ? u1 u2).
  26
  27 (* Basic constructions ******************************************************)
  28
  29 lemma subset_eq_refl (A):
  30       reflexive … (subset_eq A).
  31 /2 width=1 by conj/ qed.
  32
  33 lemma subset_eq_sym (A):
  34       symmetric … (subset_eq A).
  35 #A #u1 #u2 * /2 width=1 by conj/
  36 qed-.
  37
  38 (* Main constructions *******************************************************)
  39
  40 theorem subset_eq_trans (A):
  41         Transitive … (subset_eq A).
  42 #A #u1 #u2 * #H12 #H21 #u3 * #H23 #H32
  43 /3 width=5 by subset_le_trans, conj/
  44 qed-.