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Axiom ax-inf2 7556
Description: A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2 7557 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7555 above, using our version of Infinity ax-inf 7553 and the Axiom of Regularity ax-reg 7520. We will reference ax-inf2 7556 instead of axinf2 7555 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7553 from ax-inf2 7556 is shown by theorem axinf 7559. (Contributed by NM, 30-Aug-1993.)
Assertion
Ref Expression
ax-inf2  |-  E. x
( E. y ( y  e.  x  /\  A. z  -.  z  e.  y )  /\  A. y ( y  e.  x  ->  E. z
( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) ) )
Distinct variable group:    x, y, z, w

Detailed syntax breakdown of Axiom ax-inf2
StepHypRef Expression
1 vy . . . . . 6  set  y
2 vx . . . . . 6  set  x
31, 2wel 1722 . . . . 5  wff  y  e.  x
4 vz . . . . . . . 8  set  z
54, 1wel 1722 . . . . . . 7  wff  z  e.  y
65wn 3 . . . . . 6  wff  -.  z  e.  y
76, 4wal 1546 . . . . 5  wff  A. z  -.  z  e.  y
83, 7wa 359 . . . 4  wff  ( y  e.  x  /\  A. z  -.  z  e.  y )
98, 1wex 1547 . . 3  wff  E. y
( y  e.  x  /\  A. z  -.  z  e.  y )
104, 2wel 1722 . . . . . . 7  wff  z  e.  x
11 vw . . . . . . . . . 10  set  w
1211, 4wel 1722 . . . . . . . . 9  wff  w  e.  z
1311, 1wel 1722 . . . . . . . . . 10  wff  w  e.  y
1411, 1weq 1650 . . . . . . . . . 10  wff  w  =  y
1513, 14wo 358 . . . . . . . . 9  wff  ( w  e.  y  \/  w  =  y )
1612, 15wb 177 . . . . . . . 8  wff  ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) )
1716, 11wal 1546 . . . . . . 7  wff  A. w
( w  e.  z  <-> 
( w  e.  y  \/  w  =  y ) )
1810, 17wa 359 . . . . . 6  wff  ( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) )
1918, 4wex 1547 . . . . 5  wff  E. z
( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) )
203, 19wi 4 . . . 4  wff  ( y  e.  x  ->  E. z
( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) )
2120, 1wal 1546 . . 3  wff  A. y
( y  e.  x  ->  E. z ( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) )
229, 21wa 359 . 2  wff  ( E. y ( y  e.  x  /\  A. z  -.  z  e.  y
)  /\  A. y
( y  e.  x  ->  E. z ( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) ) )
2322, 2wex 1547 1  wff  E. x
( E. y ( y  e.  x  /\  A. z  -.  z  e.  y )  /\  A. y ( y  e.  x  ->  E. z
( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) ) )
Colors of variables: wff set class
This axiom is referenced by:  zfinf2  7557
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