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Axiom ax-inf2 7580
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 7581 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7579 above, using our version of Infinity ax-inf 7577 and the Axiom of Regularity ax-reg 7544. We will reference ax-inf2 7580 instead of axinf2 7579 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7577 from ax-inf2 7580 is shown by theorem axinf 7583. (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 1726 . . . . 5  wff  y  e.  x
4 vz . . . . . . . 8  set  z
54, 1wel 1726 . . . . . . 7  wff  z  e.  y
65wn 3 . . . . . 6  wff  -.  z  e.  y
76, 4wal 1549 . . . . 5  wff  A. z  -.  z  e.  y
83, 7wa 359 . . . 4  wff  ( y  e.  x  /\  A. z  -.  z  e.  y )
98, 1wex 1550 . . 3  wff  E. y
( y  e.  x  /\  A. z  -.  z  e.  y )
104, 2wel 1726 . . . . . . 7  wff  z  e.  x
11 vw . . . . . . . . . 10  set  w
1211, 4wel 1726 . . . . . . . . 9  wff  w  e.  z
1311, 1wel 1726 . . . . . . . . . 10  wff  w  e.  y
1411, 1weq 1653 . . . . . . . . . 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 1549 . . . . . . 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 1550 . . . . 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 1549 . . 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 1550 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  7581
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