Description: Over the base theory ax-1 -- ax-5 , the axiom of separation implies the weak emptyset axiom.
By "weak emptyset axiom", we mean the axiom asserting existence of an empty set (which can be called "the" empty set when the axiom of extensionality ax-ext is posited) provided existence of a set (the True truth constant existentially quantified over a fresh variable, extru ). This is the conclusion of bj-axnul .
Note that the weak emptyset axiom implies |- ( E. x T. -> E. y T. ) without DV conditions hence also the same statement as the weak emptyset axiom without DV conditions on x , but only on y , z .
By "axiom of separation", we mean the universal closure of ax-sep , simulated here by its instance with F. substituted for ph (and with the variable used to assert existence in the weak emptyset axiom substituted for the containing set) as the hypothesis of bj-axnul .
In particular, the axiom of existence extru and the axiom of separation together imply the emptyset axiom (and conversely, the emptyset axiom implies the axiom of existence).
Note: this theorem does not require a disjointness condition on y , z , although both axioms should be stated with all variables disjoint.
This proof only uses an instance of the axiom of separation with a bounded formula, so is valid in a constructive setting (see the CZF section in the "Intuitionistic Logic Explorer" iset.mm). (Contributed by BJ, 8-Mar-2026) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | bj-axnul.axsep | ⊢ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ⊥ ) ) | |
| Assertion | bj-axnul | ⊢ ( ∃ 𝑥 ⊤ → ∃ 𝑦 ∀ 𝑧 ∈ 𝑦 ⊥ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-axnul.axsep | ⊢ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ⊥ ) ) | |
| 2 | bj-bisimpr | ⊢ ( ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ⊥ ) ) → ( 𝑧 ∈ 𝑦 → ⊥ ) ) | |
| 3 | 2 | alimi | ⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ⊥ ) ) → ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ⊥ ) ) |
| 4 | 3 | ralrid | ⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ⊥ ) ) → ∀ 𝑧 ∈ 𝑦 ⊥ ) |
| 5 | 4 | eximi | ⊢ ( ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ⊥ ) ) → ∃ 𝑦 ∀ 𝑧 ∈ 𝑦 ⊥ ) |
| 6 | 5 1 | bj-alimii | ⊢ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∈ 𝑦 ⊥ |
| 7 | bj-spvw | ⊢ ( ∃ 𝑥 ⊤ → ( ∃ 𝑦 ∀ 𝑧 ∈ 𝑦 ⊥ ↔ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ∈ 𝑦 ⊥ ) ) | |
| 8 | 6 7 | mpbiri | ⊢ ( ∃ 𝑥 ⊤ → ∃ 𝑦 ∀ 𝑧 ∈ 𝑦 ⊥ ) |