Proof step 1 from proof sub-del-04 (conclusion) (back to tree)

⊦ s ⊆ s0 -- a ↔ s ⊆ s0 ∧ ¬ a ∈ s

System applied rule normalize

And obtained the following new goals

Proof step 2:
¬(s ⊆ s0 -- a ↔ s ⊆ s0 ∧ ¬ a ∈ s) ⊦

Rule argument

Additional information

Proof step 2 from proof sub-del-04 (previous step) (back to tree)

¬(s ⊆ s0 -- a ↔ s ⊆ s0 ∧ ¬ a ∈ s) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
⊦ (∀ a0. a0 ∈ s → a0 ≠ a ∧ a0 ∈ s0) ↔ (∀ a. a ∈ s → a ∈ s0) ∧ ¬ a ∈ s

Rule argument

Additional information

Used simplifier rules: Delete: ⊦ a ∈ s -- b ↔ a ≠ b ∧ a ∈ s; ⊦ s1 ⊆ s2 ↔ (∀ a. a ∈ s1 → a ∈ s2); (s)

Proof step 3 from proof sub-del-04 (previous step) (back to tree)

⊦ (∀ a0. a0 ∈ s → a0 ≠ a ∧ a0 ∈ s0) ↔ (∀ a. a ∈ s → a ∈ s0) ∧ ¬ a ∈ s

Heuristic cut applied rule cut formula

And obtained the following new goals

Proof step 4:
a ∈ s ⊦ (∀ a0. a0 ∈ s → a0 ≠ a ∧ a0 ∈ s0) ↔ (∀ a. a ∈ s → a ∈ s0) ∧ ¬ a ∈ s

Rule argument

fma: a ∈ s

Additional information

Used formulas: a ∈ s

Proof step 4 from proof sub-del-04 (previous step) (back to tree)

a ∈ s ⊦ (∀ a0. a0 ∈ s → a0 ≠ a ∧ a0 ∈ s0) ↔ (∀ a. a ∈ s → a ∈ s0) ∧ ¬ a ∈ s

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
a ∈ s, a0 ∈ s, ∀ a0. a0 ∈ s → a0 ≠ a ∧ a0 ∈ s0 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ a ∈ ∅; (s) ⊦ s1 = s2 ↔ (∀ a. a ∈ s1 ↔ a ∈ s2); (s) ne-01: ⊦ a ∈ s → s ≠ ∅; (forward) from specification set-basic

Proof step 5 from proof sub-del-04 (previous step) (back to tree)

a ∈ s, a0 ∈ s, ∀ a0. a0 ∈ s → a0 ≠ a ∧ a0 ∈ s0 ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: a

Additional information

Used terms: a

Proof Tree for sub-del-04 (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.