Theorem bj-axnul 37592

Description: Over the base theory ax-1 6-- ax-5 1937, 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 2741 is posited) provided existence of a set (the True truth constant existentially quantified over a fresh variable, extru 2002). This is the conclusion of bj-axnul 37592.

Note that the weak emptyset axiom implies ⊢ (∃𝑥⊤ → ∃𝑦⊤) 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 5258, simulated here by its instance with ⊥ substituted for 𝜑 (and with the variable used to assert existence in the weak emptyset axiom substituted for the containing set) as the hypothesis of bj-axnul 37592.

In particular, the axiom of existence extru 2002 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.)

Hypothesis

Ref	Expression
bj-axnul.axsep	⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥))

Assertion

Ref	Expression
bj-axnul	⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

Proof of Theorem bj-axnul

Step	Hyp	Ref	Expression
1		bj-bisimpr 37031	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → (𝑧 ∈ 𝑦 → ⊥))
2	1	alimi 1838	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → ∀𝑧(𝑧 ∈ 𝑦 → ⊥))
3	2	ralrid 3093	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → ∀𝑧 ∈ 𝑦 ⊥)
4	3	eximi 1862	. . 3 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)
5		bj-axnul.axsep	. . 3 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥))
6	4, 5	bj-alimii 37095	. 2 ⊢ ∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥
7		bj-spvw 37142	. 2 ⊢ (∃𝑥⊤ → (∃𝑦∀𝑧 ∈ 𝑦 ⊥ ↔ ∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥))
8	6, 7	mpbiri 261	1 ⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565  ⊤wtru 1568  ⊥wfal 1579  ∃wex 1806  ∀wral 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-ral 3086

This theorem is referenced by: (None)