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| Description: A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2 5965 shows it converted to abbreviations. This axiom was derived as theorem axinf2 5963 above, using our version of Infinity ax-inf 5961 and the Axiom of Regularity ax-reg 5928. We will reference ax-inf2 5964 instead of axinf2 5963 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 5961 from ax-inf2 5964 is shown by theorem axinf 5967. |
| Ref | Expression |
|---|---|
| ax-inf2 |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vy |
. . . . . . 7
| |
| 2 | 1 | cv 1585 |
. . . . . 6
|
| 3 | vx |
. . . . . . 7
| |
| 4 | 3 | cv 1585 |
. . . . . 6
|
| 5 | 2, 4 | wcel 1588 |
. . . . 5
|
| 6 | vz |
. . . . . . . . 9
| |
| 7 | 6 | cv 1585 |
. . . . . . . 8
|
| 8 | 7, 2 | wcel 1588 |
. . . . . . 7
|
| 9 | 8 | wn 2 |
. . . . . 6
|
| 10 | 9, 6 | wal 1584 |
. . . . 5
|
| 11 | 5, 10 | wa 337 |
. . . 4
|
| 12 | 11, 1 | wex 1615 |
. . 3
|
| 13 | 7, 4 | wcel 1588 |
. . . . . . 7
|
| 14 | vw |
. . . . . . . . . . 11
| |
| 15 | 14 | cv 1585 |
. . . . . . . . . 10
|
| 16 | 15, 7 | wcel 1588 |
. . . . . . . . 9
|
| 17 | 15, 2 | wcel 1588 |
. . . . . . . . . 10
|
| 18 | 15, 2 | wceq 1586 |
. . . . . . . . . 10
|
| 19 | 17, 18 | wo 336 |
. . . . . . . . 9
|
| 20 | 16, 19 | wb 219 |
. . . . . . . 8
|
| 21 | 20, 14 | wal 1584 |
. . . . . . 7
|
| 22 | 13, 21 | wa 337 |
. . . . . 6
|
| 23 | 22, 6 | wex 1615 |
. . . . 5
|
| 24 | 5, 23 | wi 3 |
. . . 4
|
| 25 | 24, 1 | wal 1584 |
. . 3
|
| 26 | 12, 25 | wa 337 |
. 2
|
| 27 | 26, 3 | wex 1615 |
1
|
| Colors of variables: wff set class |
| This axiom is referenced by: zfinf2 5965 |