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Metamath Proof Explorer

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 )