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Axiom ax-dc 8067
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 8143. 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 1623 . . . . . 6  class  y
3 vz . . . . . . 7  set  z
43cv 1623 . . . . . 6  class  z
5 vx . . . . . . 7  set  x
65cv 1623 . . . . . 6  class  x
72, 4, 6wbr 4024 . . . . 5  wff  y x z
87, 3wex 1529 . . . 4  wff  E. z 
y x z
98, 1wex 1529 . . 3  wff  E. y E. z  y x
z
106crn 4689 . . . 4  class  ran  x
116cdm 4688 . . . 4  class  dom  x
1210, 11wss 3153 . . 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 1623 . . . . . 6  class  n
16 vf . . . . . . 7  set  f
1716cv 1623 . . . . . 6  class  f
1815, 17cfv 5221 . . . . 5  class  ( f `
 n )
1915csuc 4393 . . . . . 6  class  suc  n
2019, 17cfv 5221 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 4024 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 4655 . . . 4  class  om
2321, 14, 22wral 2544 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1529 . 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  8068  axdc2lem  8069
  Copyright terms: Public domain W3C validator