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Axiom ax-dc 8282
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 8357. 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 1648 . . . . . 6  class  y
3 vz . . . . . . 7  set  z
43cv 1648 . . . . . 6  class  z
5 vx . . . . . . 7  set  x
65cv 1648 . . . . . 6  class  x
72, 4, 6wbr 4172 . . . . 5  wff  y x z
87, 3wex 1547 . . . 4  wff  E. z 
y x z
98, 1wex 1547 . . 3  wff  E. y E. z  y x
z
106crn 4838 . . . 4  class  ran  x
116cdm 4837 . . . 4  class  dom  x
1210, 11wss 3280 . . 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 1648 . . . . . 6  class  n
16 vf . . . . . . 7  set  f
1716cv 1648 . . . . . 6  class  f
1815, 17cfv 5413 . . . . 5  class  ( f `
 n )
1915csuc 4543 . . . . . 6  class  suc  n
2019, 17cfv 5413 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 4172 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 4804 . . . 4  class  om
2321, 14, 22wral 2666 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1547 . 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  8283  axdc2lem  8284
  Copyright terms: Public domain W3C validator