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Axiom ax-dc 8319
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 8394. 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 1651 . . . . . 6  class  y
3 vz . . . . . . 7  set  z
43cv 1651 . . . . . 6  class  z
5 vx . . . . . . 7  set  x
65cv 1651 . . . . . 6  class  x
72, 4, 6wbr 4205 . . . . 5  wff  y x z
87, 3wex 1550 . . . 4  wff  E. z 
y x z
98, 1wex 1550 . . 3  wff  E. y E. z  y x
z
106crn 4872 . . . 4  class  ran  x
116cdm 4871 . . . 4  class  dom  x
1210, 11wss 3313 . . 3  wff  ran  x  C_ 
dom  x
139, 12wa 359 . 2  wff  ( E. y E. z  y x z  /\  ran  x  C_  dom  x )
14 vn . . . . . . 7  set  n
1514cv 1651 . . . . . 6  class  n
16 vf . . . . . . 7  set  f
1716cv 1651 . . . . . 6  class  f
1815, 17cfv 5447 . . . . 5  class  ( f `
 n )
1915csuc 4576 . . . . . 6  class  suc  n
2019, 17cfv 5447 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 4205 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 4838 . . . 4  class  om
2321, 14, 22wral 2698 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1550 . 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  8320  axdc2lem  8321
  Copyright terms: Public domain W3C validator