Step |
Hyp |
Ref |
Expression |
1 |
|
1onn |
|- 1o e. _om |
2 |
|
ssid |
|- 1o C_ 1o |
3 |
|
ssnnfi |
|- ( ( 1o e. _om /\ 1o C_ 1o ) -> 1o e. Fin ) |
4 |
1 2 3
|
mp2an |
|- 1o e. Fin |
5 |
|
enfii |
|- ( ( 1o e. Fin /\ B ~~ 1o ) -> B e. Fin ) |
6 |
4 5
|
mpan |
|- ( B ~~ 1o -> B e. Fin ) |
7 |
6
|
adantl |
|- ( ( A e. B /\ B ~~ 1o ) -> B e. Fin ) |
8 |
|
snssi |
|- ( A e. B -> { A } C_ B ) |
9 |
8
|
adantr |
|- ( ( A e. B /\ B ~~ 1o ) -> { A } C_ B ) |
10 |
|
ensn1g |
|- ( A e. B -> { A } ~~ 1o ) |
11 |
|
ensym |
|- ( B ~~ 1o -> 1o ~~ B ) |
12 |
|
entr |
|- ( ( { A } ~~ 1o /\ 1o ~~ B ) -> { A } ~~ B ) |
13 |
10 11 12
|
syl2an |
|- ( ( A e. B /\ B ~~ 1o ) -> { A } ~~ B ) |
14 |
|
fisseneq |
|- ( ( B e. Fin /\ { A } C_ B /\ { A } ~~ B ) -> { A } = B ) |
15 |
7 9 13 14
|
syl3anc |
|- ( ( A e. B /\ B ~~ 1o ) -> { A } = B ) |
16 |
15
|
eqcomd |
|- ( ( A e. B /\ B ~~ 1o ) -> B = { A } ) |