Axiom ax-inf2 4613

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 4614 shows it converted to abbreviations. This axiom was derived as theorem axinf2 4612 above, using our version of Infinity ax-inf 4610 and the Axiom of Regularity ax-reg 4581. We will reference ax-inf2 4613 instead of axinf2 4612 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 955	. . . . . 6 class y
3		vx	. . . . . . 7 set x
4	3	cv 955	. . . . . 6 class x
5	2, 4	wcel 958	. . . . 5 wff y e. x
6		vz	. . . . . . . . 9 set z
7	6	cv 955	. . . . . . . 8 class z
8	7, 2	wcel 958	. . . . . . 7 wff z e. y
9	8	wn 2	. . . . . 6 wff -. z e. y
10	9, 6	wal 954	. . . . 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 980	. . 3 wff E.y(y e. x /\ A.z -. z e. y)
13	7, 4	wcel 958	. . . . . . 7 wff z e. x
14		vw	. . . . . . . . . . 11 set w
15	14	cv 955	. . . . . . . . . 10 class w
16	15, 7	wcel 958	. . . . . . . . 9 wff w e. z
17	15, 2	wcel 958	. . . . . . . . . 10 wff w e. y
18	15, 2	wceq 956	. . . . . . . . . 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 954	. . . . . . 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 980	. . . . 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 954	. . 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 980	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 4614

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