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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.) |
Ref | Expression |
---|---|
ax-dc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vy |
. . . . . . 7
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2 | 1 | cv 1369 |
. . . . . 6
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3 | vz |
. . . . . . 7
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4 | 3 | cv 1369 |
. . . . . 6
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5 | vx |
. . . . . . 7
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6 | 5 | cv 1369 |
. . . . . 6
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7 | 2, 4, 6 | wbr 4403 |
. . . . 5
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8 | 7, 3 | wex 1587 |
. . . 4
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9 | 8, 1 | wex 1587 |
. . 3
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10 | 6 | crn 4952 |
. . . 4
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11 | 6 | cdm 4951 |
. . . 4
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12 | 10, 11 | wss 3439 |
. . 3
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13 | 9, 12 | wa 369 |
. 2
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14 | vn |
. . . . . . 7
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15 | 14 | cv 1369 |
. . . . . 6
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16 | vf |
. . . . . . 7
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17 | 16 | cv 1369 |
. . . . . 6
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18 | 15, 17 | cfv 5529 |
. . . . 5
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19 | 15 | csuc 4832 |
. . . . . 6
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20 | 19, 17 | cfv 5529 |
. . . . 5
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21 | 18, 20, 6 | wbr 4403 |
. . . 4
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22 | com 6589 |
. . . 4
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23 | 21, 14, 22 | wral 2799 |
. . 3
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24 | 23, 16 | wex 1587 |
. 2
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25 | 13, 24 | wi 4 |
1
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Colors of variables: wff setvar class |
This axiom is referenced by: dcomex 8731 axdc2lem 8732 |
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