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Axiom ax-inf 8161
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 8145 and inf2 8146). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8165 and omex 8166 and are based on the (nontrivial) proof of inf3 8158. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8164. Theorem inf0 8144 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8168 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8164 requires this axiom along with Regularity ax-reg 8125 for its derivation (as theorem axinf2 8163 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8164 instead of this one. The derivation of this axiom from ax-inf2 8164 is shown by theorem axinf 8167.

Proofs should normally use the standard version ax-inf2 8164 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 1905 . . 3  wff  x  e.  y
4 vz . . . . . 6  setvar  z
54, 2wel 1905 . . . . 5  wff  z  e.  y
6 vw . . . . . . . 8  setvar  w
74, 6wel 1905 . . . . . . 7  wff  z  e.  w
86, 2wel 1905 . . . . . . 7  wff  w  e.  y
97, 8wa 376 . . . . . 6  wff  ( z  e.  w  /\  w  e.  y )
109, 6wex 1671 . . . . 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 1450 . . 3  wff  A. z
( z  e.  y  ->  E. w ( z  e.  w  /\  w  e.  y ) )
133, 12wa 376 . 2  wff  ( x  e.  y  /\  A. z ( z  e.  y  ->  E. w
( z  e.  w  /\  w  e.  y
) ) )
1413, 2wex 1671 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  8162
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