Theorem ominf4

Description: _om is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014) (Proof shortened by Mario Carneiro, 16-May-2015)

	Ref	Expression
Assertion	ominf4	|- -. _om e. Fin4

Proof

Step	Hyp	Ref	Expression
1		id	|- ( _om e. Fin4 -> _om e. Fin4 )
2		peano1	|- (/) e. _om
3		difsnpss	|- ( (/) e. _om <-> ( _om \ { (/) } ) C. _om )
4	2 3	mpbi	|- ( _om \ { (/) } ) C. _om
5		limom	|- Lim _om
6	5	limenpsi	|- ( _om e. Fin4 -> _om ~~ ( _om \ { (/) } ) )
7	6	ensymd	|- ( _om e. Fin4 -> ( _om \ { (/) } ) ~~ _om )
8		fin4i	|- ( ( ( _om \ { (/) } ) C. _om /\ ( _om \ { (/) } ) ~~ _om ) -> -. _om e. Fin4 )
9	4 7 8	sylancr	|- ( _om e. Fin4 -> -. _om e. Fin4 )
10	1 9	pm2.65i	|- -. _om e. Fin4