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Axiom ax-dc 8829
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 8904. 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 1382 . . . . . 6  class  y
3 vz . . . . . . 7  setvar  z
43cv 1382 . . . . . 6  class  z
5 vx . . . . . . 7  setvar  x
65cv 1382 . . . . . 6  class  x
72, 4, 6wbr 4437 . . . . 5  wff  y x z
87, 3wex 1599 . . . 4  wff  E. z 
y x z
98, 1wex 1599 . . 3  wff  E. y E. z  y x
z
106crn 4990 . . . 4  class  ran  x
116cdm 4989 . . . 4  class  dom  x
1210, 11wss 3461 . . 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 1382 . . . . . 6  class  n
16 vf . . . . . . 7  setvar  f
1716cv 1382 . . . . . 6  class  f
1815, 17cfv 5578 . . . . 5  class  ( f `
 n )
1915csuc 4870 . . . . . 6  class  suc  n
2019, 17cfv 5578 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 4437 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 6685 . . . 4  class  om
2321, 14, 22wral 2793 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1599 . 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  8830  axdc2lem  8831
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