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

Axiom ax-inf2 7310
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 7311 shows it converted to abbreviations. This axiom was derived as theorem axinf2 7309 above, using our version of Infinity ax-inf 7307 and the Axiom of Regularity ax-reg 7274. We will reference ax-inf2 7310 instead of axinf2 7309 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 7307 from ax-inf2 7310 is shown by theorem axinf 7313. (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 1622 . . . . 5  wff  y  e.  x
4 vz . . . . . . . 8  set  z
54, 1wel 1622 . . . . . . 7  wff  z  e.  y
65wn 5 . . . . . 6  wff  -.  z  e.  y
76, 4wal 1532 . . . . 5  wff  A. z  -.  z  e.  y
83, 7wa 360 . . . 4  wff  ( y  e.  x  /\  A. z  -.  z  e.  y )
98, 1wex 1537 . . 3  wff  E. y
( y  e.  x  /\  A. z  -.  z  e.  y )
104, 2wel 1622 . . . . . . 7  wff  z  e.  x
11 vw . . . . . . . . . 10  set  w
1211, 4wel 1622 . . . . . . . . 9  wff  w  e.  z
1311, 1wel 1622 . . . . . . . . . 10  wff  w  e.  y
1411, 1weq 1620 . . . . . . . . . 10  wff  w  =  y
1513, 14wo 359 . . . . . . . . 9  wff  ( w  e.  y  \/  w  =  y )
1612, 15wb 178 . . . . . . . 8  wff  ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) )
1716, 11wal 1532 . . . . . . 7  wff  A. w
( w  e.  z  <-> 
( w  e.  y  \/  w  =  y ) )
1810, 17wa 360 . . . . . 6  wff  ( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) )
1918, 4wex 1537 . . . . 5  wff  E. z
( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) )
203, 19wi 6 . . . 4  wff  ( y  e.  x  ->  E. z
( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) )
2120, 1wal 1532 . . 3  wff  A. y
( y  e.  x  ->  E. z ( z  e.  x  /\  A. w ( w  e.  z  <->  ( w  e.  y  \/  w  =  y ) ) ) )
229, 21wa 360 . 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 1537 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  7311
  Copyright terms: Public domain W3C validator