Step |
Hyp |
Ref |
Expression |
1 |
|
pwpr |
|- ~P { X , Y } = ( { (/) , { X } } u. { { Y } , { X , Y } } ) |
2 |
1
|
eleq2i |
|- ( P e. ~P { X , Y } <-> P e. ( { (/) , { X } } u. { { Y } , { X , Y } } ) ) |
3 |
|
elun |
|- ( P e. ( { (/) , { X } } u. { { Y } , { X , Y } } ) <-> ( P e. { (/) , { X } } \/ P e. { { Y } , { X , Y } } ) ) |
4 |
2 3
|
bitri |
|- ( P e. ~P { X , Y } <-> ( P e. { (/) , { X } } \/ P e. { { Y } , { X , Y } } ) ) |
5 |
|
fveq2 |
|- ( P = (/) -> ( # ` P ) = ( # ` (/) ) ) |
6 |
|
hash0 |
|- ( # ` (/) ) = 0 |
7 |
6
|
eqeq2i |
|- ( ( # ` P ) = ( # ` (/) ) <-> ( # ` P ) = 0 ) |
8 |
|
eqeq1 |
|- ( ( # ` P ) = 0 -> ( ( # ` P ) = 2 <-> 0 = 2 ) ) |
9 |
|
0ne2 |
|- 0 =/= 2 |
10 |
|
eqneqall |
|- ( 0 = 2 -> ( 0 =/= 2 -> P = { X , Y } ) ) |
11 |
9 10
|
mpi |
|- ( 0 = 2 -> P = { X , Y } ) |
12 |
8 11
|
syl6bi |
|- ( ( # ` P ) = 0 -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
13 |
7 12
|
sylbi |
|- ( ( # ` P ) = ( # ` (/) ) -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
14 |
5 13
|
syl |
|- ( P = (/) -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
15 |
|
hashsng |
|- ( X e. _V -> ( # ` { X } ) = 1 ) |
16 |
|
fveq2 |
|- ( { X } = P -> ( # ` { X } ) = ( # ` P ) ) |
17 |
16
|
eqcoms |
|- ( P = { X } -> ( # ` { X } ) = ( # ` P ) ) |
18 |
17
|
eqeq1d |
|- ( P = { X } -> ( ( # ` { X } ) = 1 <-> ( # ` P ) = 1 ) ) |
19 |
|
eqeq1 |
|- ( ( # ` P ) = 1 -> ( ( # ` P ) = 2 <-> 1 = 2 ) ) |
20 |
|
1ne2 |
|- 1 =/= 2 |
21 |
|
eqneqall |
|- ( 1 = 2 -> ( 1 =/= 2 -> P = { X , Y } ) ) |
22 |
20 21
|
mpi |
|- ( 1 = 2 -> P = { X , Y } ) |
23 |
19 22
|
syl6bi |
|- ( ( # ` P ) = 1 -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
24 |
18 23
|
syl6bi |
|- ( P = { X } -> ( ( # ` { X } ) = 1 -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) ) |
25 |
15 24
|
syl5com |
|- ( X e. _V -> ( P = { X } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) ) |
26 |
|
snprc |
|- ( -. X e. _V <-> { X } = (/) ) |
27 |
|
eqeq2 |
|- ( { X } = (/) -> ( P = { X } <-> P = (/) ) ) |
28 |
5 6
|
eqtrdi |
|- ( P = (/) -> ( # ` P ) = 0 ) |
29 |
28
|
eqeq1d |
|- ( P = (/) -> ( ( # ` P ) = 2 <-> 0 = 2 ) ) |
30 |
29 11
|
syl6bi |
|- ( P = (/) -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
31 |
27 30
|
syl6bi |
|- ( { X } = (/) -> ( P = { X } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) ) |
32 |
26 31
|
sylbi |
|- ( -. X e. _V -> ( P = { X } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) ) |
33 |
25 32
|
pm2.61i |
|- ( P = { X } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
34 |
14 33
|
jaoi |
|- ( ( P = (/) \/ P = { X } ) -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
35 |
|
hashsng |
|- ( Y e. _V -> ( # ` { Y } ) = 1 ) |
36 |
|
fveq2 |
|- ( { Y } = P -> ( # ` { Y } ) = ( # ` P ) ) |
37 |
36
|
eqcoms |
|- ( P = { Y } -> ( # ` { Y } ) = ( # ` P ) ) |
38 |
37
|
eqeq1d |
|- ( P = { Y } -> ( ( # ` { Y } ) = 1 <-> ( # ` P ) = 1 ) ) |
39 |
38 23
|
syl6bi |
|- ( P = { Y } -> ( ( # ` { Y } ) = 1 -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) ) |
40 |
35 39
|
syl5com |
|- ( Y e. _V -> ( P = { Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) ) |
41 |
|
snprc |
|- ( -. Y e. _V <-> { Y } = (/) ) |
42 |
|
eqeq2 |
|- ( { Y } = (/) -> ( P = { Y } <-> P = (/) ) ) |
43 |
5
|
eqeq1d |
|- ( P = (/) -> ( ( # ` P ) = 2 <-> ( # ` (/) ) = 2 ) ) |
44 |
6
|
eqeq1i |
|- ( ( # ` (/) ) = 2 <-> 0 = 2 ) |
45 |
44 11
|
sylbi |
|- ( ( # ` (/) ) = 2 -> P = { X , Y } ) |
46 |
43 45
|
syl6bi |
|- ( P = (/) -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
47 |
42 46
|
syl6bi |
|- ( { Y } = (/) -> ( P = { Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) ) |
48 |
41 47
|
sylbi |
|- ( -. Y e. _V -> ( P = { Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) ) |
49 |
40 48
|
pm2.61i |
|- ( P = { Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
50 |
|
ax-1 |
|- ( P = { X , Y } -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
51 |
49 50
|
jaoi |
|- ( ( P = { Y } \/ P = { X , Y } ) -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
52 |
34 51
|
jaoi |
|- ( ( ( P = (/) \/ P = { X } ) \/ ( P = { Y } \/ P = { X , Y } ) ) -> ( ( # ` P ) = 2 -> P = { X , Y } ) ) |
53 |
|
elpri |
|- ( P e. { (/) , { X } } -> ( P = (/) \/ P = { X } ) ) |
54 |
|
elpri |
|- ( P e. { { Y } , { X , Y } } -> ( P = { Y } \/ P = { X , Y } ) ) |
55 |
53 54
|
orim12i |
|- ( ( P e. { (/) , { X } } \/ P e. { { Y } , { X , Y } } ) -> ( ( P = (/) \/ P = { X } ) \/ ( P = { Y } \/ P = { X , Y } ) ) ) |
56 |
52 55
|
syl11 |
|- ( ( # ` P ) = 2 -> ( ( P e. { (/) , { X } } \/ P e. { { Y } , { X , Y } } ) -> P = { X , Y } ) ) |
57 |
4 56
|
syl5bi |
|- ( ( # ` P ) = 2 -> ( P e. ~P { X , Y } -> P = { X , Y } ) ) |
58 |
57
|
imp |
|- ( ( ( # ` P ) = 2 /\ P e. ~P { X , Y } ) -> P = { X , Y } ) |