Axiom ax-inf2 8890

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 8891 shows it converted to abbreviations. This axiom was derived as theorem axinf2 8889 above, using our version of Infinity ax-inf 8887 and the Axiom of Regularity ax-reg 8843. We will reference ax-inf2 8890 instead of axinf2 8889 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 8887 from ax-inf2 8890 is shown by theorem axinf 8893. (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 2049	. . . . 5 wff 𝑦 ∈ 𝑥
4		vz	. . . . . . . 8 setvar 𝑧
5	4, 1	wel 2049	. . . . . . 7 wff 𝑧 ∈ 𝑦
6	5	wn 3	. . . . . 6 wff ¬ 𝑧 ∈ 𝑦
7	6, 4	wal 1505	. . . . 5 wff ∀𝑧 ¬ 𝑧 ∈ 𝑦
8	3, 7	wa 387	. . . 4 wff (𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)
9	8, 1	wex 1742	. . 3 wff ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)
10	4, 2	wel 2049	. . . . . . 7 wff 𝑧 ∈ 𝑥
11		vw	. . . . . . . . . 10 setvar 𝑤
12	11, 4	wel 2049	. . . . . . . . 9 wff 𝑤 ∈ 𝑧
13	11, 1	wel 2049	. . . . . . . . . 10 wff 𝑤 ∈ 𝑦
14	11, 1	weq 1922	. . . . . . . . . 10 wff 𝑤 = 𝑦
15	13, 14	wo 833	. . . . . . . . 9 wff (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)
16	12, 15	wb 198	. . . . . . . 8 wff (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
17	16, 11	wal 1505	. . . . . . 7 wff ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
18	10, 17	wa 387	. . . . . 6 wff (𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))
19	18, 4	wex 1742	. . . . 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 1505	. . 3 wff ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
22	9, 21	wa 387	. 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 1742	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:  zfinf2  8891