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Proof step 9 from proof intu-empty (previous step)

(∀ a. ¬ a ∈ s ∨ ¬ a ∈ s0) ∧ (∀ a. ¬ a ∈ s ∨ ¬ a ∈ s1) ⊦ ∀ a. ¬ a ∈ s ∨ ¬ a ∈ s0 ∪ s1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
a ∈ s, a ∈ s0 ∨ a ∈ s1, ∀ a. ¬ a ∈ s ∨ ¬ a ∈ s1, ∀ a. ¬ a ∈ s ∨ ¬ a ∈ s0 ⊦ s = ∅, # s = 0

Rule argument

Additional information

Used simplifier rules: unrax: ⊦ a ∈ s ∪ s0 ↔ ¬ ¬(a ∈ s ∨ a ∈ s0);%una: ⊦ (s ∪ s0) ∪ s1 = s ∪ s0 ∪ s1;%unc: ⊦ s ∪ s0 = s0 ∪ s;%size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic%ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic