Axiom ax-inf2 4597

Description: A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf 4598 shows it converted to abbreviations. This axiom was derived as theorem axinf2 4596 above, using our version of Infinity ax-inf 4594 and the Axiom of Regularity ax-reg 4565. We will reference ax-inf2 4597 instead of axinf2 4596 so that the ordinary uses of Regularity can be more easily identified.

Assertion

Ref	Expression
ax-inf2	⊢ ∃x(∃y(y ∈ x ⋀ ∀z ¬ z ∈ y) ⋀ ∀y(y ∈ x → ∃z(z ∈ x ⋀ ∀w(w ∈ z ↔ (w ∈ y ⋁ w = y)))))

Distinct variable group:   x,y,z,w

Detailed syntax breakdown of Axiom ax-inf2

Step	Hyp	Ref	Expression
1		vy	. . . . . . 7 set y
2	1	cv 952	. . . . . 6 class y
3		vx	. . . . . . 7 set x
4	3	cv 952	. . . . . 6 class x
5	2, 4	wcel 955	. . . . 5 wff y ∈ x
6		vz	. . . . . . . . 9 set z
7	6	cv 952	. . . . . . . 8 class z
8	7, 2	wcel 955	. . . . . . 7 wff z ∈ y
9	8	wn 2	. . . . . 6 wff ¬ z ∈ y
10	9, 6	wal 951	. . . . 5 wff ∀z ¬ z ∈ y
11	5, 10	wa 223	. . . 4 wff (y ∈ x ⋀ ∀z ¬ z ∈ y)
12	11, 1	wex 977	. . 3 wff ∃y(y ∈ x ⋀ ∀z ¬ z ∈ y)
13	7, 4	wcel 955	. . . . . . 7 wff z ∈ x
14		vw	. . . . . . . . . . 11 set w
15	14	cv 952	. . . . . . . . . 10 class w
16	15, 7	wcel 955	. . . . . . . . 9 wff w ∈ z
17	15, 2	wcel 955	. . . . . . . . . 10 wff w ∈ y
18	15, 2	wceq 953	. . . . . . . . . 10 wff w = y
19	17, 18	wo 222	. . . . . . . . 9 wff (w ∈ y ⋁ w = y)
20	16, 19	wb 146	. . . . . . . 8 wff (w ∈ z ↔ (w ∈ y ⋁ w = y))
21	20, 14	wal 951	. . . . . . 7 wff ∀w(w ∈ z ↔ (w ∈ y ⋁ w = y))
22	13, 21	wa 223	. . . . . 6 wff (z ∈ x ⋀ ∀w(w ∈ z ↔ (w ∈ y ⋁ w = y)))
23	22, 6	wex 977	. . . . 5 wff ∃z(z ∈ x ⋀ ∀w(w ∈ z ↔ (w ∈ y ⋁ w = y)))
24	5, 23	wi 3	. . . 4 wff (y ∈ x → ∃z(z ∈ x ⋀ ∀w(w ∈ z ↔ (w ∈ y ⋁ w = y))))
25	24, 1	wal 951	. . 3 wff ∀y(y ∈ x → ∃z(z ∈ x ⋀ ∀w(w ∈ z ↔ (w ∈ y ⋁ w = y))))
26	12, 25	wa 223	. 2 wff (∃y(y ∈ x ⋀ ∀z ¬ z ∈ y) ⋀ ∀y(y ∈ x → ∃z(z ∈ x ⋀ ∀w(w ∈ z ↔ (w ∈ y ⋁ w = y)))))
27	26, 3	wex 977	1 wff ∃x(∃y(y ∈ x ⋀ ∀z ¬ z ∈ y) ⋀ ∀y(y ∈ x → ∃z(z ∈ x ⋀ ∀w(w ∈ z ↔ (w ∈ y ⋁ w = y)))))

This axiom is referenced by:  zfinf 4598

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