Step |
Hyp |
Ref |
Expression |
1 |
|
tgldim0.g |
|- P = ( E ` F ) |
2 |
|
tgldim0.p |
|- ( ph -> ( # ` P ) = 1 ) |
3 |
|
tgldim0.a |
|- ( ph -> A e. P ) |
4 |
|
tgldim0.b |
|- ( ph -> B e. P ) |
5 |
1
|
fvexi |
|- P e. _V |
6 |
|
hash1snb |
|- ( P e. _V -> ( ( # ` P ) = 1 <-> E. x P = { x } ) ) |
7 |
5 6
|
ax-mp |
|- ( ( # ` P ) = 1 <-> E. x P = { x } ) |
8 |
2 7
|
sylib |
|- ( ph -> E. x P = { x } ) |
9 |
3
|
adantr |
|- ( ( ph /\ P = { x } ) -> A e. P ) |
10 |
|
simpr |
|- ( ( ph /\ P = { x } ) -> P = { x } ) |
11 |
9 10
|
eleqtrd |
|- ( ( ph /\ P = { x } ) -> A e. { x } ) |
12 |
|
elsni |
|- ( A e. { x } -> A = x ) |
13 |
11 12
|
syl |
|- ( ( ph /\ P = { x } ) -> A = x ) |
14 |
4
|
adantr |
|- ( ( ph /\ P = { x } ) -> B e. P ) |
15 |
14 10
|
eleqtrd |
|- ( ( ph /\ P = { x } ) -> B e. { x } ) |
16 |
|
elsni |
|- ( B e. { x } -> B = x ) |
17 |
15 16
|
syl |
|- ( ( ph /\ P = { x } ) -> B = x ) |
18 |
13 17
|
eqtr4d |
|- ( ( ph /\ P = { x } ) -> A = B ) |
19 |
8 18
|
exlimddv |
|- ( ph -> A = B ) |