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Axiom ax-dc 8088
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 8164. 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 1631 . . . . . 6  class  y
3 vz . . . . . . 7  set  z
43cv 1631 . . . . . 6  class  z
5 vx . . . . . . 7  set  x
65cv 1631 . . . . . 6  class  x
72, 4, 6wbr 4039 . . . . 5  wff  y x z
87, 3wex 1531 . . . 4  wff  E. z 
y x z
98, 1wex 1531 . . 3  wff  E. y E. z  y x
z
106crn 4706 . . . 4  class  ran  x
116cdm 4705 . . . 4  class  dom  x
1210, 11wss 3165 . . 3  wff  ran  x  C_ 
dom  x
139, 12wa 358 . 2  wff  ( E. y E. z  y x z  /\  ran  x  C_  dom  x )
14 vn . . . . . . 7  set  n
1514cv 1631 . . . . . 6  class  n
16 vf . . . . . . 7  set  f
1716cv 1631 . . . . . 6  class  f
1815, 17cfv 5271 . . . . 5  class  ( f `
 n )
1915csuc 4410 . . . . . 6  class  suc  n
2019, 17cfv 5271 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 4039 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 4672 . . . 4  class  om
2321, 14, 22wral 2556 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1531 . 2  wff  E. f A. n  e.  om  ( f `  n
) x ( f `
 suc  n )
2513, 24wi 4 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  8089  axdc2lem  8090
  Copyright terms: Public domain W3C validator