Axiom ax-inf2 9656

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 zfinf2 9657 shows it converted to abbreviations. This axiom was derived as Theorem axinf2 9655 above, using our version of Infinity ax-inf 9653 and the Axiom of Regularity ax-reg 9600. We will reference ax-inf2 9656 instead of axinf2 9655 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 9653 from ax-inf2 9656 is shown by Theorem axinf 9659. (Contributed by NM, 3-Nov-1996.)

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	. . . . . 6 setvar 𝑦
2		vx	. . . . . 6 setvar 𝑥
3	1, 2	wel 2147	. . . . 5 wff 𝑦 ∈ 𝑥
4		vz	. . . . . . . 8 setvar 𝑧
5	4, 1	wel 2147	. . . . . . 7 wff 𝑧 ∈ 𝑦
6	5	wn 3	. . . . . 6 wff ¬ 𝑧 ∈ 𝑦
7	6, 4	wal 1567	. . . . 5 wff ∀𝑧 ¬ 𝑧 ∈ 𝑦
8	3, 7	wa 401	. . . 4 wff (𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)
9	8, 1	wex 1811	. . 3 wff ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)
10	4, 2	wel 2147	. . . . . . 7 wff 𝑧 ∈ 𝑥
11		vw	. . . . . . . . . 10 setvar 𝑤
12	11, 4	wel 2147	. . . . . . . . 9 wff 𝑤 ∈ 𝑧
13	11, 1	wel 2147	. . . . . . . . . 10 wff 𝑤 ∈ 𝑦
14	11, 1	weq 1996	. . . . . . . . . 10 wff 𝑤 = 𝑦
15	13, 14	wo 861	. . . . . . . . 9 wff (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)
16	12, 15	wb 209	. . . . . . . 8 wff (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
17	16, 11	wal 1567	. . . . . . 7 wff ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
18	10, 17	wa 401	. . . . . 6 wff (𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))
19	18, 4	wex 1811	. . . . 5 wff ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))
20	3, 19	wi 4	. . . 4 wff (𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
21	20, 1	wal 1567	. . 3 wff ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
22	9, 21	wa 401	. 2 wff (∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
23	22, 2	wex 1811	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 used by:  zfinf2  9657  axnulALT3  35798  axprALT2  35799