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Axiom ax-inf 7947
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 7931 and inf2 7932). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 7951 and omex 7952 and are based on the (nontrivial) proof of inf3 7944. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 7950. Theorem inf0 7930 shows the reverse derivation of our axiom from a standard one. Theorem inf5 7954 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 7950 requires this axiom along with Regularity ax-reg 7910 for its derivation (as theorem axinf2 7949 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 7950 instead of this one. The derivation of this axiom from ax-inf2 7950 is shown by theorem axinf 7953.

Proofs should normally use the standard version ax-inf2 7950 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 1759 . . 3  wff  x  e.  y
4 vz . . . . . 6  setvar  z
54, 2wel 1759 . . . . 5  wff  z  e.  y
6 vw . . . . . . . 8  setvar  w
74, 6wel 1759 . . . . . . 7  wff  z  e.  w
86, 2wel 1759 . . . . . . 7  wff  w  e.  y
97, 8wa 369 . . . . . 6  wff  ( z  e.  w  /\  w  e.  y )
109, 6wex 1587 . . . . 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 1368 . . 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 1587 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  7948
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