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Step * of Lemma fset-minimals-ac-le

∀[T:Type]. ∀eq:EqDecider(T). ∀s:fset(fset(T)).  fset-ac-le(eq;s;fset-minimals(xs,ys.f-proper-subset-dec(eq;xs;ys); s))
BY
{ (Auto
   THEN Unfold `fset-ac-le` 0
   THEN (InstLemma `fset-all-iff` [⌜fset(T)⌝;⌜deq-fset(eq)⌝]⋅ THENA Auto)
   THEN RWO  "-1" 0
   THEN Auto
   THEN (RW assert_pushdownC 0 THENA Auto)
   THEN RWO "assert-fset-null" 0
   THEN Auto) }

1
1. T : Type
2. eq : EqDecider(T)
3. s : fset(fset(T))
4. ∀[P:fset(T) ⟶ 𝔹]. ∀[s:fset(fset(T))].  uiff(fset-all(s;x.P[x]);∀[x:fset(T)]. ↑P[x] supposing x ∈ s)
5. x : fset(T)
6. x ∈ s
⊢ ¬({y ∈ fset-minimals(xs,ys.f-proper-subset-dec(eq;xs;ys); s) | deq-f-subset(eq) y x} = {} ∈ fset(fset(T)))

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Latex:

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Latex:
\mforall{}[T:Type]
    \mforall{}eq:EqDecider(T).  \mforall{}s:fset(fset(T)).
        fset-ac-le(eq;s;fset-minimals(xs,ys.f-proper-subset-dec(eq;xs;ys);  s))


By

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Latex:
(Auto
  THEN  Unfold  `fset-ac-le`  0
  THEN  (InstLemma  `fset-all-iff`  [\mkleeneopen{}fset(T)\mkleeneclose{};\mkleeneopen{}deq-fset(eq)\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  RWO    "-1"  0
  THEN  Auto
  THEN  (RW  assert\_pushdownC  0  THENA  Auto)
  THEN  RWO  "assert-fset-null"  0
  THEN  Auto)

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