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Axiom ax-inf 8044
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 8028 and inf2 8029). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8048 and omex 8049 and are based on the (nontrivial) proof of inf3 8041. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8047. Theorem inf0 8027 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8051 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8047 requires this axiom along with Regularity ax-reg 8007 for its derivation (as theorem axinf2 8046 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8047 instead of this one. The derivation of this axiom from ax-inf2 8047 is shown by theorem axinf 8050.

Proofs should normally use the standard version ax-inf2 8047 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 1763 . . 3  wff  x  e.  y
4 vz . . . . . 6  setvar  z
54, 2wel 1763 . . . . 5  wff  z  e.  y
6 vw . . . . . . . 8  setvar  w
74, 6wel 1763 . . . . . . 7  wff  z  e.  w
86, 2wel 1763 . . . . . . 7  wff  w  e.  y
97, 8wa 369 . . . . . 6  wff  ( z  e.  w  /\  w  e.  y )
109, 6wex 1591 . . . . 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 1372 . . 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 1591 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  8045
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