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Axiom ax-inf2 7342
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 7343 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7341 above, using our version of Infinity ax-inf 7339 and the Axiom of Regularity ax-reg 7306. We will reference ax-inf2 7342 instead of axinf2 7341 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7339 from ax-inf2 7342 is shown by theorem axinf 7345. (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 1685 . . . . 5  wff  y  e.  x
4 vz . . . . . . . 8  set  z
54, 1wel 1685 . . . . . . 7  wff  z  e.  y
65wn 3 . . . . . 6  wff  -.  z  e.  y
76, 4wal 1527 . . . . 5  wff  A. z  -.  z  e.  y
83, 7wa 358 . . . 4  wff  ( y  e.  x  /\  A. z  -.  z  e.  y )
98, 1wex 1528 . . 3  wff  E. y
( y  e.  x  /\  A. z  -.  z  e.  y )
104, 2wel 1685 . . . . . . 7  wff  z  e.  x
11 vw . . . . . . . . . 10  set  w
1211, 4wel 1685 . . . . . . . . 9  wff  w  e.  z
1311, 1wel 1685 . . . . . . . . . 10  wff  w  e.  y
1411, 1weq 1624 . . . . . . . . . 10  wff  w  =  y
1513, 14wo 357 . . . . . . . . 9  wff  ( w  e.  y  \/  w  =  y )
1612, 15wb 176 . . . . . . . 8  wff  ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) )
1716, 11wal 1527 . . . . . . 7  wff  A. w
( w  e.  z  <-> 
( w  e.  y  \/  w  =  y ) )
1810, 17wa 358 . . . . . 6  wff  ( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) )
1918, 4wex 1528 . . . . 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 1527 . . 3  wff  A. y
( y  e.  x  ->  E. z ( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) )
229, 21wa 358 . 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 1528 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  7343
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