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Proof step 3 from proof keys-inodekey-in (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 4:
∀ ns0, n. # ns0 < # ns → true → (inodekey(n) ∈ ns0.keys ↔ n ∈ ns0) ⊦ inodekey(n) ∈ ns.keys ↔ n ∈ ns, ns = ∅

Rule argument

fma: ns = ∅

Additional information

Used simplifier rules: keys-base: ⊦ ∅.keys = ∅;% ⊦ ¬ key ∈ ∅; (s)% ⊦ ¬ n ∈ ∅; (s) Used formulas: ns = ∅