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Axiom ax-inf 7272
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 7256 and inf2 7257). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 7276 and omex 7277 and are based on the (nontrivial) proof of inf3 7269. Our version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 7275. Theorem inf0 7255 shows the reverse derivation of our axiom from a standard one. Theorem inf5 7279 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 7275 requires this axiom along with Regularity ax-reg 7239 for its derivation (as theorem axinf2 7274 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 7275 instead of this one. The derivation of this axiom from ax-inf2 7275 is shown by theorem axinf 7278. (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  set  x
2 vy . . . 4  set  y
31, 2wel 1622 . . 3  wff  x  e.  y
4 vz . . . . . 6  set  z
54, 2wel 1622 . . . . 5  wff  z  e.  y
6 vw . . . . . . . 8  set  w
74, 6wel 1622 . . . . . . 7  wff  z  e.  w
86, 2wel 1622 . . . . . . 7  wff  w  e.  y
97, 8wa 360 . . . . . 6  wff  ( z  e.  w  /\  w  e.  y )
109, 6wex 1537 . . . . 5  wff  E. w
( z  e.  w  /\  w  e.  y
)
115, 10wi 6 . . . 4  wff  ( z  e.  y  ->  E. w
( z  e.  w  /\  w  e.  y
) )
1211, 4wal 1532 . . 3  wff  A. z
( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y ) )
133, 12wa 360 . 2  wff  ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w
( z  e.  w  /\  w  e.  y
) ) )
1413, 2wex 1537 1  wff  E. y
( x  e.  y  /\  A. z ( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y ) ) )
Colors of variables: wff set class
This axiom is referenced by:  zfinf  7273
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