Axiom ax-inf2 4634

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 4635 shows it converted to abbreviations. This axiom was derived as theorem axinf2 4633 above, using our version of Infinity ax-inf 4631 and the Axiom of Regularity ax-reg 4602. We will reference ax-inf2 4634 instead of axinf2 4633 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 957	. . . . . 6 class y
3		vx	. . . . . . 7 set x
4	3	cv 957	. . . . . 6 class x
5	2, 4	wcel 960	. . . . 5 wff y e. x
6		vz	. . . . . . . . 9 set z
7	6	cv 957	. . . . . . . 8 class z
8	7, 2	wcel 960	. . . . . . 7 wff z e. y
9	8	wn 2	. . . . . 6 wff -. z e. y
10	9, 6	wal 956	. . . . 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 982	. . 3 wff E.y(y e. x /\ A.z -. z e. y)
13	7, 4	wcel 960	. . . . . . 7 wff z e. x
14		vw	. . . . . . . . . . 11 set w
15	14	cv 957	. . . . . . . . . 10 class w
16	15, 7	wcel 960	. . . . . . . . 9 wff w e. z
17	15, 2	wcel 960	. . . . . . . . . 10 wff w e. y
18	15, 2	wceq 958	. . . . . . . . . 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 956	. . . . . . 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 982	. . . . 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 956	. . 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 982	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 4635

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