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Axiom ax-inf 8143
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 8127 and inf2 8128). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8147 and omex 8148 and are based on the (nontrivial) proof of inf3 8140. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8146. Theorem inf0 8126 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8150 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8146 requires this axiom along with Regularity ax-reg 8107 for its derivation (as theorem axinf2 8145 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8146 instead of this one. The derivation of this axiom from ax-inf2 8146 is shown by theorem axinf 8149.

Proofs should normally use the standard version ax-inf2 8146 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 1888 . . 3  wff  x  e.  y
4 vz . . . . . 6  setvar  z
54, 2wel 1888 . . . . 5  wff  z  e.  y
6 vw . . . . . . . 8  setvar  w
74, 6wel 1888 . . . . . . 7  wff  z  e.  w
86, 2wel 1888 . . . . . . 7  wff  w  e.  y
97, 8wa 371 . . . . . 6  wff  ( z  e.  w  /\  w  e.  y )
109, 6wex 1663 . . . . 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 1442 . . 3  wff  A. z
( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y ) )
133, 12wa 371 . 2  wff  ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w
( z  e.  w  /\  w  e.  y
) ) )
1413, 2wex 1663 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  8144
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