Theorem bj-axnul 37518

Description: Over the base theory ax-1 6-- ax-5 1929, 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 2733 is posited) provided existence of a set (the True truth constant existentially quantified over a fresh variable, extru 1994). This is the conclusion of bj-axnul 37518.

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 5243, 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 37518.

In particular, the axiom of existence extru 1994 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 36957	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → (𝑧 ∈ 𝑦 → ⊥))
2	1	alimi 1830	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → ∀𝑧(𝑧 ∈ 𝑦 → ⊥))
3	2	ralrid 3083	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) → ∀𝑧 ∈ 𝑦 ⊥)
4	3	eximi 1854	. . 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 37021	. 2 ⊢ ∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥
7		bj-spvw 37068	. 2 ⊢ (∃𝑥⊤ → (∃𝑦∀𝑧 ∈ 𝑦 ⊥ ↔ ∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥))
8	6, 7	mpbiri 260	1 ⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557  ⊤wtru 1560  ⊥wfal 1571  ∃wex 1798  ∀wral 3075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-ral 3076

This theorem is referenced by: (None)