Axiom ax-inf2 4549

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 4550 shows it converted to abbreviations. This axiom was derived as theorem axinf2 4548 above, using our version of Infinity ax-inf 4546 and the Axiom of Regularity ax-reg 4517. We will reference ax-inf2 4549 instead of axinf2 4548 so that the ordinary uses of Regularity can be more easily identified.

Assertion

Ref	Expression
ax-inf2	|- E.x(E.y(y e. x /\ A.z -. z e. y) /\ A.y(y e. x -> E.z(z e. x /\ A.w(w e. z <-> (w e. 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 1098	. . . . . 6 class y
3		vx	. . . . . . 7 set x
4	3	cv 1098	. . . . . 6 class x
5	2, 4	wcel 1105	. . . . 5 wff y e. x
6		vz	. . . . . . . . 9 set z
7	6	cv 1098	. . . . . . . 8 class z
8	7, 2	wcel 1105	. . . . . . 7 wff z e. y
9	8	wn 2	. . . . . 6 wff -. z e. y
10	9, 6	wal 950	. . . . 5 wff A.z -. z e. y
11	5, 10	wa 223	. . . 4 wff (y e. x /\ A.z -. z e. y)
12	11, 1	wex 956	. . 3 wff E.y(y e. x /\ A.z -. z e. y)
13	7, 4	wcel 1105	. . . . . . 7 wff z e. x
14		vw	. . . . . . . . . . 11 set w
15	14	cv 1098	. . . . . . . . . 10 class w
16	15, 7	wcel 1105	. . . . . . . . 9 wff w e. z
17	15, 2	wcel 1105	. . . . . . . . . 10 wff w e. y
18	15, 2	wceq 1099	. . . . . . . . . 10 wff w = y
19	17, 18	wo 222	. . . . . . . . 9 wff (w e. y \/ w = y)
20	16, 19	wb 146	. . . . . . . 8 wff (w e. z <-> (w e. y \/ w = y))
21	20, 14	wal 950	. . . . . . 7 wff A.w(w e. z <-> (w e. y \/ w = y))
22	13, 21	wa 223	. . . . . 6 wff (z e. x /\ A.w(w e. z <-> (w e. y \/ w = y)))
23	22, 6	wex 956	. . . . 5 wff E.z(z e. x /\ A.w(w e. z <-> (w e. y \/ w = y)))
24	5, 23	wi 3	. . . 4 wff (y e. x -> E.z(z e. x /\ A.w(w e. z <-> (w e. y \/ w = y))))
25	24, 1	wal 950	. . 3 wff A.y(y e. x -> E.z(z e. x /\ A.w(w e. z <-> (w e. y \/ w = y))))
26	12, 25	wa 223	. 2 wff (E.y(y e. x /\ A.z -. z e. y) /\ A.y(y e. x -> E.z(z e. x /\ A.w(w e. z <-> (w e. y \/ w = y)))))
27	26, 3	wex 956	1 wff E.x(E.y(y e. x /\ A.z -. z e. y) /\ A.y(y e. x -> E.z(z e. x /\ A.w(w e. z <-> (w e. y \/ w = y)))))

This axiom is referenced by:  zfinf 4550

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