Theorem bnj529

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj529.1	|- D = ( _om \ { (/) } )
Assertion	bnj529	|- ( M e. D -> (/) e. M )

Proof

Step	Hyp	Ref	Expression
1		bnj529.1	|- D = ( _om \ { (/) } )
2		eldifsn	|- ( M e. ( _om \ { (/) } ) <-> ( M e. _om /\ M =/= (/) ) )
3	2	biimpi	|- ( M e. ( _om \ { (/) } ) -> ( M e. _om /\ M =/= (/) ) )
4	3 1	eleq2s	|- ( M e. D -> ( M e. _om /\ M =/= (/) ) )
5		nnord	|- ( M e. _om -> Ord M )
6	5	anim1i	|- ( ( M e. _om /\ M =/= (/) ) -> ( Ord M /\ M =/= (/) ) )
7		ord0eln0	|- ( Ord M -> ( (/) e. M <-> M =/= (/) ) )
8	7	biimpar	|- ( ( Ord M /\ M =/= (/) ) -> (/) e. M )
9	4 6 8	3syl	|- ( M e. D -> (/) e. M )