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Proof step 8 from proof rem-inter-empty3 (previous step)

(s0 \ s1) ∩ s = ∅, a ∈ s2, a ∈ s, ∀ a0. ¬ a0 ∈ s2 \ s0 ∨ ¬ a0 ∈ s ⊦ s2 = ∅, s = ∅, #((s2 \ s1) ∩ s) = 0, # s2 = 0, # s = 0, a ∈ s1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2)
Proof step 10:
∀ a. a ∈ (s0 \ s1) ∩ s ↔ a ∈ ∅, a ∈ s2, a ∈ s, ∀ a0. ¬ a0 ∈ s2 \ s0 ∨ ¬ a0 ∈ s ⊦ s2 = ∅, s = ∅, #((s2 \ s1) ∩ s) = 0, # s2 = 0, # s = 0, a ∈ s1

Rule argument

spec: set-basic inst: name: Extension seq: ⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2) subst: [s2, s1] → [∅, (s0 \ s1) ∩ s] flags: rewrite

Additional information

Used terms: ∅, (s0 \ s1) ∩ s