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

Axiom ax-dc 8026
Description: Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. Dependent choice is equivalent to the statement that every (nonempty) pruned tree has a branch. This axiom is redundant in ZFC; see axdc 8102. But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013.)
Assertion
Ref Expression
ax-dc  |-  ( ( E. y E. z 
y x z  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n
) )
Distinct variable group:    f, n, x, y, z

Detailed syntax breakdown of Axiom ax-dc
StepHypRef Expression
1 vy . . . . . . 7  set  y
21cv 1618 . . . . . 6  class  y
3 vz . . . . . . 7  set  z
43cv 1618 . . . . . 6  class  z
5 vx . . . . . . 7  set  x
65cv 1618 . . . . . 6  class  x
72, 4, 6wbr 3983 . . . . 5  wff  y x z
87, 3wex 1537 . . . 4  wff  E. z 
y x z
98, 1wex 1537 . . 3  wff  E. y E. z  y x
z
106crn 4648 . . . 4  class  ran  x
116cdm 4647 . . . 4  class  dom  x
1210, 11wss 3113 . . 3  wff  ran  x  C_ 
dom  x
139, 12wa 360 . 2  wff  ( E. y E. z  y x z  /\  ran  x  C_  dom  x )
14 vn . . . . . . 7  set  n
1514cv 1618 . . . . . 6  class  n
16 vf . . . . . . 7  set  f
1716cv 1618 . . . . . 6  class  f
1815, 17cfv 4659 . . . . 5  class  ( f `
 n )
1915csuc 4352 . . . . . 6  class  suc  n
2019, 17cfv 4659 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 3983 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 4614 . . . 4  class  om
2321, 14, 22wral 2516 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1537 . 2  wff  E. f A. n  e.  om  ( f `  n
) x ( f `
 suc  n )
2513, 24wi 6 1  wff  ( ( E. y E. z 
y x z  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n
) )
Colors of variables: wff set class
This axiom is referenced by:  dcomex  8027  axdc2lem  8028
  Copyright terms: Public domain W3C validator