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Proof step 7 from proof keys-inodekey-in (previous step)

∀ ns0, n. # ns0 < # ns → true → (inodekey(n) ∈ ns0.keys ↔ n ∈ ns0) ⊦ # ns = 0, inodekey(n) ∈ ns.keys ↔ n ∈ ns, ∀ n. n ∈ ns ↔ n ∈ ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
n0 ∈ ns, ∀ ns0, n. # ns0 < # ns → true → (inodekey(n) ∈ ns0.keys ↔ n ∈ ns0) ⊦ # ns = 0, inodekey(n) ∈ ns.keys ↔ n ∈ ns

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ n ∈ ∅; (s)