Theorem bnj1001 31028

Description: Technical lemma for bnj69 31078. 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.)

Hypotheses

Ref	Expression
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

Ref	Expression
bnj1001	|- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. om /\ suc i e. p ) )

Proof of Theorem bnj1001

Step	Hyp	Ref	Expression
1		bnj1001.27	. 2 |- ( ( th /\ ch /\ ta /\ et ) -> ch" )
2		bnj1001.6	. . . . 5 |- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
3	2	simplbi 476	. . . 4 |- ( et -> i e. n )
4	3	bnj708 30826	. . 3 |- ( ( th /\ ch /\ ta /\ et ) -> i e. n )
5		bnj1001.3	. . . . . 6 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
6	5	bnj1232 30874	. . . . 5 |- ( ch -> n e. D )
7	6	bnj706 30824	. . . 4 |- ( ( th /\ ch /\ ta /\ et ) -> n e. D )
8		bnj1001.13	. . . . 5 |- D = ( om \ { (/) } )
9	8	bnj923 30838	. . . 4 |- ( n e. D -> n e. om )
10	7, 9	syl 17	. . 3 |- ( ( th /\ ch /\ ta /\ et ) -> n e. om )
11		elnn 7075	. . 3 |- ( ( i e. n /\ n e. om ) -> i e. om )
12	4, 10, 11	syl2anc 693	. 2 |- ( ( th /\ ch /\ ta /\ et ) -> i e. om )
13		bnj1001.5	. . . . . 6 |- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
14	13	simp3bi 1078	. . . . 5 |- ( ta -> p = suc n )
15	14	bnj707 30825	. . . 4 |- ( ( th /\ ch /\ ta /\ et ) -> p = suc n )
16		nnord 7073	. . . . . . 7 |- ( n e. om -> Ord n )
17		ordsucelsuc 7022	. . . . . . 7 |- ( Ord n -> ( i e. n <-> suc i e. suc n ) )
18	9, 16, 17	3syl 18	. . . . . 6 |- ( n e. D -> ( i e. n <-> suc i e. suc n ) )
19	18	biimpa 501	. . . . 5 |- ( ( n e. D /\ i e. n ) -> suc i e. suc n )
20		eleq2 2690	. . . . 5 |- ( p = suc n -> ( suc i e. p <-> suc i e. suc n ) )
21	19, 20	anim12i 590	. . . 4 |- ( ( ( n e. D /\ i e. n ) /\ p = suc n ) -> ( suc i e. suc n /\ ( suc i e. p <-> suc i e. suc n ) ) )
22	7, 4, 15, 21	syl21anc 1325	. . 3 |- ( ( th /\ ch /\ ta /\ et ) -> ( suc i e. suc n /\ ( suc i e. p <-> suc i e. suc n ) ) )
23		bianir 1009	. . 3 |- ( ( suc i e. suc n /\ ( suc i e. p <-> suc i e. suc n ) ) -> suc i e. p )
24	22, 23	syl 17	. 2 |- ( ( th /\ ch /\ ta /\ et ) -> suc i e. p )
25	1, 12, 24	3jca 1242	1 |- ( ( th /\ ch /\ ta /\ et ) -> ( ch" /\ i e. om /\ suc i e. p ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   (/)c0 3915   {csn 4177   Ord word 5722   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj1020  31033