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Proof step 9 from proof unsub-union-01 (previous step)

a ∈ s, a1 ∈ s2, a ∈ s2, a ∈ s1, a2 ∈ s1, ∀ a. a ∈ s1 ∨ a ∈ s2 → a ∈ s ⊦ a ∈ s ∧ (a0 ∈ s2 → a0 ∈ s)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
a0 ∈ s2, a2 ∈ s1, a ∈ s1, a ∈ s2, a1 ∈ s2, a ∈ s, ∀ a. a ∈ s1 ∨ a ∈ s2 → a ∈ s ⊦ s = ∅, # s = 0, a0 ∈ s

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic%ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic