MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-inf2 Structured version   Unicode version

Axiom ax-inf2 7596
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 7597 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7595 above, using our version of Infinity ax-inf 7593 and the Axiom of Regularity ax-reg 7560. We will reference ax-inf2 7596 instead of axinf2 7595 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7593 from ax-inf2 7596 is shown by theorem axinf 7599. (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  7597
  Copyright terms: Public domain W3C validator