Axiom ax-inf2 9624

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 9625 shows it converted to abbreviations. This axiom was derived as Theorem axinf2 9623 above, using our version of Infinity ax-inf 9621 and the Axiom of Regularity ax-reg 9568. We will reference ax-inf2 9624 instead of axinf2 9623 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 9621 from ax-inf2 9624 is shown by Theorem axinf 9627. (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 2146	. . . . 5 wff 𝑦 ∈ 𝑥
4		vz	. . . . . . . 8 setvar 𝑧
5	4, 1	wel 2146	. . . . . . 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 401	. . . 4 wff (𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)
9	8, 1	wex 1812	. . 3 wff ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦)
10	4, 2	wel 2146	. . . . . . 7 wff 𝑧 ∈ 𝑥
11		vw	. . . . . . . . . 10 setvar 𝑤
12	11, 4	wel 2146	. . . . . . . . 9 wff 𝑤 ∈ 𝑧
13	11, 1	wel 2146	. . . . . . . . . 10 wff 𝑤 ∈ 𝑦
14	11, 1	weq 1995	. . . . . . . . . 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 1568	. . . . . . 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 1812	. . . . 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 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 1812	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  9625  axnulALT3  35624  axprALT2  35625