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Axiom ax-inf 7582
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 7566 and inf2 7567). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 7586 and omex 7587 and are based on the (nontrivial) proof of inf3 7579. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 7585. Theorem inf0 7565 shows the reverse derivation of our axiom from a standard one. Theorem inf5 7589 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 7585 requires this axiom along with Regularity ax-reg 7549 for its derivation (as theorem axinf2 7584 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 7585 instead of this one. The derivation of this axiom from ax-inf2 7585 is shown by theorem axinf 7588.

Proofs should normally use the standard version ax-inf2 7585 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  set  x
2 vy . . . 4  set  y
31, 2wel 1726 . . 3  wff  x  e.  y
4 vz . . . . . 6  set  z
54, 2wel 1726 . . . . 5  wff  z  e.  y
6 vw . . . . . . . 8  set  w
74, 6wel 1726 . . . . . . 7  wff  z  e.  w
86, 2wel 1726 . . . . . . 7  wff  w  e.  y
97, 8wa 359 . . . . . 6  wff  ( z  e.  w  /\  w  e.  y )
109, 6wex 1550 . . . . 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 1549 . . 3  wff  A. z
( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y ) )
133, 12wa 359 . 2  wff  ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w
( z  e.  w  /\  w  e.  y
) ) )
1413, 2wex 1550 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  7583
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