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Axiom ax-dc 8605
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 8680. 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 1363 . . . . . 6  class  y
3 vz . . . . . . 7  setvar  z
43cv 1363 . . . . . 6  class  z
5 vx . . . . . . 7  setvar  x
65cv 1363 . . . . . 6  class  x
72, 4, 6wbr 4282 . . . . 5  wff  y x z
87, 3wex 1591 . . . 4  wff  E. z 
y x z
98, 1wex 1591 . . 3  wff  E. y E. z  y x
z
106crn 4830 . . . 4  class  ran  x
116cdm 4829 . . . 4  class  dom  x
1210, 11wss 3318 . . 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 1363 . . . . . 6  class  n
16 vf . . . . . . 7  setvar  f
1716cv 1363 . . . . . 6  class  f
1815, 17cfv 5408 . . . . 5  class  ( f `
 n )
1915csuc 4710 . . . . . 6  class  suc  n
2019, 17cfv 5408 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 4282 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 6467 . . . 4  class  om
2321, 14, 22wral 2707 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1591 . 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  8606  axdc2lem  8607
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