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Axiom ax-inf 8058
Description: Axiom of Infinity. An axiom of Zermelo-Fraenkel set theory. This axiom is the gateway to "Cantor's paradise" (an expression coined by Hilbert). It asserts that given a starting set  x, an infinite set  y built from it exists. Although our version is apparently not given in the literature, it is similar to, but slightly shorter than, the Axiom of Infinity in [FreydScedrov] p. 283 (see inf1 8042 and inf2 8043). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8062 and omex 8063 and are based on the (nontrivial) proof of inf3 8055. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8061. Theorem inf0 8041 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8065 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8061 requires this axiom along with Regularity ax-reg 8021 for its derivation (as theorem axinf2 8060 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8061 instead of this one. The derivation of this axiom from ax-inf2 8061 is shown by theorem axinf 8064.

Proofs should normally use the standard version ax-inf2 8061 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

Assertion
Ref Expression
ax-inf  |-  E. y
( x  e.  y  /\  A. z ( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y ) ) )
Distinct variable group:    x, y, z, w

Detailed syntax breakdown of Axiom ax-inf
StepHypRef Expression
1 vx . . . 4  setvar  x
2 vy . . . 4  setvar  y
31, 2wel 1805 . . 3  wff  x  e.  y
4 vz . . . . . 6  setvar  z
54, 2wel 1805 . . . . 5  wff  z  e.  y
6 vw . . . . . . . 8  setvar  w
74, 6wel 1805 . . . . . . 7  wff  z  e.  w
86, 2wel 1805 . . . . . . 7  wff  w  e.  y
97, 8wa 369 . . . . . 6  wff  ( z  e.  w  /\  w  e.  y )
109, 6wex 1599 . . . . 5  wff  E. w
( z  e.  w  /\  w  e.  y
)
115, 10wi 4 . . . 4  wff  ( z  e.  y  ->  E. w
( z  e.  w  /\  w  e.  y
) )
1211, 4wal 1381 . . 3  wff  A. z
( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y ) )
133, 12wa 369 . 2  wff  ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w
( z  e.  w  /\  w  e.  y
) ) )
1413, 2wex 1599 1  wff  E. y
( x  e.  y  /\  A. z ( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y ) ) )
Colors of variables: wff setvar class
This axiom is referenced by:  zfinf  8059
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