Metamath Proof Explorer

Theorem bnj1001

Description: Technical lemma for bnj69 . 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 bnj1001.3 
|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1001.5 
|- ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
bnj1001.6 
|- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
bnj1001.13 
|- D = ( _om \ { (/) } )
bnj1001.27 
|- ( ( th /\ ch /\ ta /\ et ) -> ch" )
Assertion bnj1001 
|- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. _om /\ suc i e. p ) )

Proof

Step Hyp Ref Expression
1 bnj1001.3 
 |-  ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2 bnj1001.5 
 |-  ( ta <-> ( m e. _om /\ n = suc m /\ p = suc n ) )
3 bnj1001.6 
 |-  ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
4 bnj1001.13 
 |-  D = ( _om \ { (/) } )
5 bnj1001.27 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ch" )
6 3 simplbi 
 |-  ( et -> i e. n )
7 6 bnj708 
 |-  ( ( th /\ ch /\ ta /\ et ) -> i e. n )
8 1 bnj1232 
 |-  ( ch -> n e. D )
9 8 bnj706 
 |-  ( ( th /\ ch /\ ta /\ et ) -> n e. D )
10 4 bnj923 
 |-  ( n e. D -> n e. _om )
11 9 10 syl 
 |-  ( ( th /\ ch /\ ta /\ et ) -> n e. _om )
12 elnn 
 |-  ( ( i e. n /\ n e. _om ) -> i e. _om )
13 7 11 12 syl2anc 
 |-  ( ( th /\ ch /\ ta /\ et ) -> i e. _om )
14 2 simp3bi 
 |-  ( ta -> p = suc n )
15 14 bnj707 
 |-  ( ( th /\ ch /\ ta /\ et ) -> p = suc n )
16 nnord 
 |-  ( n e. _om -> Ord n )
17 ordsucelsuc 
 |-  ( Ord n -> ( i e. n <-> suc i e. suc n ) )
18 10 16 17 3syl 
 |-  ( n e. D -> ( i e. n <-> suc i e. suc n ) )
19 18 biimpa 
 |-  ( ( n e. D /\ i e. n ) -> suc i e. suc n )
20 eleq2 
 |-  ( p = suc n -> ( suc i e. p <-> suc i e. suc n ) )
21 19 20 anim12i 
 |-  ( ( ( n e. D /\ i e. n ) /\ p = suc n ) -> ( suc i e. suc n /\ ( suc i e. p <-> suc i e. suc n ) ) )
22 9 7 15 21 syl21anc 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( suc i e. suc n /\ ( suc i e. p <-> suc i e. suc n ) ) )
23 bianir 
 |-  ( ( suc i e. suc n /\ ( suc i e. p <-> suc i e. suc n ) ) -> suc i e. p )
24 22 23 syl 
 |-  ( ( th /\ ch /\ ta /\ et ) -> suc i e. p )
25 5 13 24 3jca 
 |-  ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. _om /\ suc i e. p ) )