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Axiom ax-dc 8730
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 8805. 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 1369 . . . . . 6  class  y
3 vz . . . . . . 7  setvar  z
43cv 1369 . . . . . 6  class  z
5 vx . . . . . . 7  setvar  x
65cv 1369 . . . . . 6  class  x
72, 4, 6wbr 4403 . . . . 5  wff  y x z
87, 3wex 1587 . . . 4  wff  E. z 
y x z
98, 1wex 1587 . . 3  wff  E. y E. z  y x
z
106crn 4952 . . . 4  class  ran  x
116cdm 4951 . . . 4  class  dom  x
1210, 11wss 3439 . . 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 1369 . . . . . 6  class  n
16 vf . . . . . . 7  setvar  f
1716cv 1369 . . . . . 6  class  f
1815, 17cfv 5529 . . . . 5  class  ( f `
 n )
1915csuc 4832 . . . . . 6  class  suc  n
2019, 17cfv 5529 . . . . 5  class  ( f `
 suc  n )
2118, 20, 6wbr 4403 . . . 4  wff  ( f `
 n ) x ( f `  suc  n )
22 com 6589 . . . 4  class  om
2321, 14, 22wral 2799 . . 3  wff  A. n  e.  om  ( f `  n ) x ( f `  suc  n
)
2423, 16wex 1587 . 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  8731  axdc2lem  8732
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