Axiom ax-regs 35505

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 35518, but this derivation relies on ax-inf2 9609 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 1807	. 2 wff ∃𝑥𝜑
4		vy	. . . . . . 7 setvar 𝑦
5	2, 4	weq 1990	. . . . . 6 wff 𝑥 = 𝑦
6	5, 1	wi 4	. . . . 5 wff (𝑥 = 𝑦 → 𝜑)
7	6, 2	wal 1566	. . . 4 wff ∀𝑥(𝑥 = 𝑦 → 𝜑)
8		vz	. . . . . . 7 setvar 𝑧
9	8, 4	wel 2142	. . . . . 6 wff 𝑧 ∈ 𝑦
10	2, 8	weq 1990	. . . . . . . . 9 wff 𝑥 = 𝑧
11	10, 1	wi 4	. . . . . . . 8 wff (𝑥 = 𝑧 → 𝜑)
12	11, 2	wal 1566	. . . . . . 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 1566	. . . 4 wff ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))
16	7, 15	wa 400	. . 3 wff (∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑)))
17	16, 4	wex 1807	. 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  35506  axregscl  35507