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Axiom ax-dc 8072
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 8148. 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 1622 . . . . . 6  class  y
3 vz . . . . . . 7  set  z
43cv 1622 . . . . . 6  class  z
5 vx . . . . . . 7  set  x
65cv 1622 . . . . . 6  class  x
72, 4, 6wbr 4023 . . . . 5  wff  y x z
87, 3wex 1528 . . . 4  wff  E. z 
y x z
98, 1wex 1528 . . 3  wff  E. y E. z  y x
z
106crn 4690 . . . 4  class  ran  x
116cdm 4689 . . . 4  class  dom  x
1210, 11wss 3152 . . 3  wff  ran  x  C_ 
dom  x
139, 12wa 358 . 2  wff  ( E. y E. z  y x z  /\  ran  x  C_  dom  x )
14 vn . . . . . . 7  set  n
1514cv 1622 . . . . . 6  class  n
16 vf . . . . . . 7  set  f
1716cv 1622 . . . . . 6  class  f
1815, 17cfv 5255 . . . . 5  class  ( f `
 n )
1915csuc 4394 . . . . . 6  class  suc  n
2019, 17cfv 5255 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 4023 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 4656 . . . 4  class  om
2321, 14, 22wral 2543 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1528 . 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  8073  axdc2lem  8074
  Copyright terms: Public domain W3C validator