Axiom ax-inf2 4642

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 4643 shows it converted to abbreviations. This axiom was derived as theorem axinf2 4641 above, using our version of Infinity ax-inf 4639 and the Axiom of Regularity ax-reg 4608. We will reference ax-inf2 4642 instead of axinf2 4641 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 959	. . . . . 6 class y
3		vx	. . . . . . 7 set x
4	3	cv 959	. . . . . 6 class x
5	2, 4	wcel 962	. . . . 5 wff y ∈ x
6		vz	. . . . . . . . 9 set z
7	6	cv 959	. . . . . . . 8 class z
8	7, 2	wcel 962	. . . . . . 7 wff z ∈ y
9	8	wn 2	. . . . . 6 wff ¬ z ∈ y
10	9, 6	wal 958	. . . . 5 wff ∀z ¬ z ∈ y
11	5, 10	wa 223	. . . 4 wff (y ∈ x ⋀ ∀z ¬ z ∈ y)
12	11, 1	wex 984	. . 3 wff ∃y(y ∈ x ⋀ ∀z ¬ z ∈ y)
13	7, 4	wcel 962	. . . . . . 7 wff z ∈ x
14		vw	. . . . . . . . . . 11 set w
15	14	cv 959	. . . . . . . . . 10 class w
16	15, 7	wcel 962	. . . . . . . . 9 wff w ∈ z
17	15, 2	wcel 962	. . . . . . . . . 10 wff w ∈ y
18	15, 2	wceq 960	. . . . . . . . . 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 958	. . . . . . 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 984	. . . . 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 958	. . 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 984	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 4643

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