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Axiom ax-dc 8836
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 8911. 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  setvar  y
21cv 1378 . . . . . 6  class  y
3 vz . . . . . . 7  setvar  z
43cv 1378 . . . . . 6  class  z
5 vx . . . . . . 7  setvar  x
65cv 1378 . . . . . 6  class  x
72, 4, 6wbr 4452 . . . . 5  wff  y x z
87, 3wex 1596 . . . 4  wff  E. z 
y x z
98, 1wex 1596 . . 3  wff  E. y E. z  y x
z
106crn 5005 . . . 4  class  ran  x
116cdm 5004 . . . 4  class  dom  x
1210, 11wss 3481 . . 3  wff  ran  x  C_ 
dom  x
139, 12wa 369 . 2  wff  ( E. y E. z  y x z  /\  ran  x  C_  dom  x )
14 vn . . . . . . 7  setvar  n
1514cv 1378 . . . . . 6  class  n
16 vf . . . . . . 7  setvar  f
1716cv 1378 . . . . . 6  class  f
1815, 17cfv 5593 . . . . 5  class  ( f `
 n )
1915csuc 4885 . . . . . 6  class  suc  n
2019, 17cfv 5593 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 4452 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 6694 . . . 4  class  om
2321, 14, 22wral 2817 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1596 . 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 setvar class
This axiom is referenced by:  dcomex  8837  axdc2lem  8838
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