Theorem bnj556

Description: Technical lemma for bnj852 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj556.18	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
	bnj556.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
Assertion	bnj556	|- ( et -> si )

Proof

Step	Hyp	Ref	Expression
1		bnj556.18	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
2		bnj556.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) )
3		vex	|- p e. _V
4	3	bnj216	|- ( m = suc p -> p e. m )
5	4	3anim3i	|- ( ( m e. D /\ n = suc m /\ m = suc p ) -> ( m e. D /\ n = suc m /\ p e. m ) )
6	5	adantr	|- ( ( ( m e. D /\ n = suc m /\ m = suc p ) /\ p e. _om ) -> ( m e. D /\ n = suc m /\ p e. m ) )
7		bnj258	|- ( ( m e. D /\ n = suc m /\ p e. _om /\ m = suc p ) <-> ( ( m e. D /\ n = suc m /\ m = suc p ) /\ p e. _om ) )
8	2 7	bitri	|- ( et <-> ( ( m e. D /\ n = suc m /\ m = suc p ) /\ p e. _om ) )
9	6 8 1	3imtr4i	|- ( et -> si )