Detailed syntax breakdown of Definition df-gzpow
Step | Hyp | Ref
| Expression |
1 | | cgzp 31345 |
. 2
class
AxPow |
2 | | c1o 7553 |
. . . . . . . 8
class
1𝑜 |
3 | | c2o 7554 |
. . . . . . . 8
class
2𝑜 |
4 | | cgoe 31315 |
. . . . . . . 8
class
∈𝑔 |
5 | 2, 3, 4 | co 6650 |
. . . . . . 7
class
(1𝑜∈𝑔2𝑜) |
6 | | c0 3915 |
. . . . . . . 8
class
∅ |
7 | 2, 6, 4 | co 6650 |
. . . . . . 7
class
(1𝑜∈𝑔∅) |
8 | | cgob 31332 |
. . . . . . 7
class
↔𝑔 |
9 | 5, 7, 8 | co 6650 |
. . . . . 6
class
((1𝑜∈𝑔2𝑜)
↔𝑔
(1𝑜∈𝑔∅)) |
10 | 9, 2 | cgol 31317 |
. . . . 5
class
∀𝑔1𝑜((1𝑜∈𝑔2𝑜)
↔𝑔 (1𝑜∈𝑔∅)) |
11 | 3, 2, 4 | co 6650 |
. . . . 5
class
(2𝑜∈𝑔1𝑜) |
12 | | cgoi 31330 |
. . . . 5
class
→𝑔 |
13 | 10, 11, 12 | co 6650 |
. . . 4
class
(∀𝑔1𝑜((1𝑜∈𝑔2𝑜)
↔𝑔 (1𝑜∈𝑔∅)) →𝑔
(2𝑜∈𝑔1𝑜)) |
14 | 13, 3 | cgol 31317 |
. . 3
class
∀𝑔2𝑜(∀𝑔1𝑜((1𝑜∈𝑔2𝑜)
↔𝑔 (1𝑜∈𝑔∅)) →𝑔
(2𝑜∈𝑔1𝑜)) |
15 | 14, 2 | cgox 31334 |
. 2
class
∃𝑔1𝑜∀𝑔2𝑜(∀𝑔1𝑜((1𝑜∈𝑔2𝑜)
↔𝑔 (1𝑜∈𝑔∅)) →𝑔 (2𝑜∈𝑔1𝑜)) |
16 | 1, 15 | wceq 1483 |
1
wff AxPow =
∃𝑔1𝑜∀𝑔2𝑜(∀𝑔1𝑜((1𝑜∈𝑔2𝑜)
↔𝑔 (1𝑜∈𝑔∅)) →𝑔 (2𝑜∈𝑔1𝑜)) |