Axiom ax-inf2 9606

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 9607 shows it converted to abbreviations. This axiom was derived as Theorem axinf2 9605 above, using our version of Infinity ax-inf 9603 and the Axiom of Regularity ax-reg 9550. We will reference ax-inf2 9606 instead of axinf2 9605 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 9603 from ax-inf2 9606 is shown by Theorem axinf 9609. (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 2144	. . . . 5 wff 𝑦 ∈ 𝑥
4		vz	. . . . . . . 8 setvar 𝑧
5	4, 1	wel 2144	. . . . . . 7 wff 𝑧 ∈ 𝑦
6	5	wn 3	. . . . . 6 wff ¬ 𝑧 ∈ 𝑦
7	6, 4	wal 1568	. . . . 5 wff ∀𝑧 ¬ 𝑧 ∈ 𝑦
8	3, 7	wa 400	. . . 4 wff (𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)
9	8, 1	wex 1809	. . 3 wff ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)
10	4, 2	wel 2144	. . . . . . 7 wff 𝑧 ∈ 𝑥
11		vw	. . . . . . . . . 10 setvar 𝑤
12	11, 4	wel 2144	. . . . . . . . 9 wff 𝑤 ∈ 𝑧
13	11, 1	wel 2144	. . . . . . . . . 10 wff 𝑤 ∈ 𝑦
14	11, 1	weq 1992	. . . . . . . . . 10 wff 𝑤 = 𝑦
15	13, 14	wo 860	. . . . . . . . 9 wff (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)
16	12, 15	wb 209	. . . . . . . 8 wff (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
17	16, 11	wal 1568	. . . . . . 7 wff ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
18	10, 17	wa 400	. . . . . 6 wff (𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))
19	18, 4	wex 1809	. . . . 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 1568	. . 3 wff ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
22	9, 21	wa 400	. 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 1809	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  9607  axnulALT3  35511  axprALT2  35512