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Theorem bnj1001 29771
Description: Technical lemma for bnj69 29821. 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 462 . . . 4  |-  ( et 
->  i  e.  n
)
43bnj708 29568 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  n )
5 bnj1001.3 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
65bnj1232 29617 . . . . 5  |-  ( ch 
->  n  e.  D
)
76bnj706 29566 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  D )
8 bnj1001.13 . . . . 5  |-  D  =  ( om  \  { (/)
} )
98bnj923 29581 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
107, 9syl 17 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  om )
11 elnn 6714 . . 3  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
124, 10, 11syl2anc 666 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  om )
13 bnj1001.5 . . . . . 6  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
1413simp3bi 1023 . . . . 5  |-  ( ta 
->  p  =  suc  n )
1514bnj707 29567 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  p  =  suc  n )
16 nnord 6712 . . . . . . 7  |-  ( n  e.  om  ->  Ord  n )
17 ordsucelsuc 6661 . . . . . . 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 487 . . . . 5  |-  ( ( n  e.  D  /\  i  e.  n )  ->  suc  i  e.  suc  n )
20 eleq2 2496 . . . . 5  |-  ( p  =  suc  n  -> 
( suc  i  e.  p 
<->  suc  i  e.  suc  n ) )
2119, 20anim12i 569 . . . 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 1264 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e. 
suc  n ) ) )
23 bianir 976 . . 3  |-  ( ( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e.  suc  n ) )  ->  suc  i  e.  p
)
2422, 23syl 17 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  suc  i  e.  p
)
251, 12, 243jca 1186 1  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ch"  /\  i  e. 
om  /\  suc  i  e.  p ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    \ cdif 3434   (/)c0 3762   {csn 3997   Ord word 5439   suc csuc 5442    Fn wfn 5594   ` cfv 5599   omcom 6704    /\ w-bnj17 29493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-tr 4517  df-eprel 4762  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-om 6705  df-bnj17 29494
This theorem is referenced by:  bnj1020  29776
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