Axiom ax-regs 35426

Description: A strong version of the Axiom of Regularity. It states that if there exists a set with property 𝜑, then there must exist a set with property 𝜑 such that none of its elements have property 𝜑. This axiom can be derived from the axioms of ZF set theory as shown in axregs 35439, but this derivation relies on ax-inf2 9594 and is thus not possible in a finitist context. (Contributed by BTernaryTau, 29-Dec-2025.)

Assertion

Ref	Expression
ax-regs	⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

Distinct variable groups:   𝜑,𝑦,𝑧   𝑥,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Axiom ax-regs

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	wex 1800	. 2 wff ∃𝑥𝜑
4		vy	. . . . . . 7 setvar 𝑦
5	2, 4	weq 1983	. . . . . 6 wff 𝑥 = 𝑦
6	5, 1	wi 4	. . . . 5 wff (𝑥 = 𝑦 → 𝜑)
7	6, 2	wal 1559	. . . 4 wff ∀𝑥(𝑥 = 𝑦 → 𝜑)
8		vz	. . . . . . 7 setvar 𝑧
9	8, 4	wel 2144	. . . . . 6 wff 𝑧 ∈ 𝑦
10	2, 8	weq 1983	. . . . . . . . 9 wff 𝑥 = 𝑧
11	10, 1	wi 4	. . . . . . . 8 wff (𝑥 = 𝑧 → 𝜑)
12	11, 2	wal 1559	. . . . . . 7 wff ∀𝑥(𝑥 = 𝑧 → 𝜑)
13	12	wn 3	. . . . . 6 wff ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)
14	9, 13	wi 4	. . . . 5 wff (𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
15	14, 8	wal 1559	. . . 4 wff ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
16	7, 15	wa 399	. . 3 wff (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
17	16, 4	wex 1800	. 2 wff ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
18	3, 17	wi 4	1 wff (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

This axiom is referenced by:  axreg  35427  axregscl  35428