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Theorem bnj1001 28990
Description: Technical lemma for bnj69 29040. 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
StepHypRef Expression
1 bnj1001.27 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  ch" )
2 bnj1001.6 . . . . 5  |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i ) ) )
32simplbi 446 . . . 4  |-  ( et 
->  i  e.  n
)
43bnj708 28785 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  n )
5 bnj1001.3 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
65bnj1232 28836 . . . . 5  |-  ( ch 
->  n  e.  D
)
76bnj706 28783 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  D )
8 bnj1001.13 . . . . 5  |-  D  =  ( om  \  { (/)
} )
98bnj923 28798 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
107, 9syl 15 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  om )
11 elnn 4666 . . 3  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
124, 10, 11syl2anc 642 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  om )
13 bnj1001.5 . . . . . 6  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
1413simp3bi 972 . . . . 5  |-  ( ta 
->  p  =  suc  n )
1514bnj707 28784 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  p  =  suc  n )
16 nnord 4664 . . . . . . 7  |-  ( n  e.  om  ->  Ord  n )
17 ordsucelsuc 4613 . . . . . . 7  |-  ( Ord  n  ->  ( i  e.  n  <->  suc  i  e.  suc  n ) )
189, 16, 173syl 18 . . . . . 6  |-  ( n  e.  D  ->  (
i  e.  n  <->  suc  i  e. 
suc  n ) )
1918biimpa 470 . . . . 5  |-  ( ( n  e.  D  /\  i  e.  n )  ->  suc  i  e.  suc  n )
20 eleq2 2344 . . . . 5  |-  ( p  =  suc  n  -> 
( suc  i  e.  p 
<->  suc  i  e.  suc  n ) )
2119, 20anim12i 549 . . . 4  |-  ( ( ( n  e.  D  /\  i  e.  n
)  /\  p  =  suc  n )  ->  ( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e.  suc  n ) ) )
227, 4, 15, 21syl21anc 1181 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e. 
suc  n ) ) )
23 bnj926 28799 . . 3  |-  ( ( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e.  suc  n ) )  ->  suc  i  e.  p
)
2422, 23syl 15 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  suc  i  e.  p
)
251, 12, 243jca 1132 1  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ch"  /\  i  e. 
om  /\  suc  i  e.  p ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149   (/)c0 3455   {csn 3640   Ord word 4391   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711
This theorem is referenced by:  bnj1020  28995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-bnj17 28712
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