Theorem bj-axnul 37425

Description: Over the base theory ax-1 6-- ax-5 1917, 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 2711 is posited) provided existence of a set (the True truth constant existentially quantified over a fresh variable, extru 1982). This is the conclusion of bj-axnul 37425.

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 5218, 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 37425.

In particular, the axiom of existence extru 1982 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 36864	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → (𝑧 ∈ 𝑦 → ⊥))
2	1	alimi 1818	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → ∀𝑧(𝑧 ∈ 𝑦 → ⊥))
3	2	ralrid 3061	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → ∀𝑧 ∈ 𝑦 ⊥)
4	3	eximi 1842	. . 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 36928	. 2 ⊢ ∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥
7		bj-spvw 36975	. 2 ⊢ (∃𝑥⊤ → (∃𝑦∀𝑧 ∈ 𝑦 ⊥ ↔ ∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥))
8	6, 7	mpbiri 259	1 ⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545  ⊤wtru 1548  ⊥wfal 1559  ∃wex 1786  ∀wral 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-ral 3054

This theorem is referenced by: (None)