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Theorem bnj1001 29035
Description: Technical lemma for bnj69 29085. 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 447 . . . 4  |-  ( et 
->  i  e.  n
)
43bnj708 28830 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  n )
5 bnj1001.3 . . . . . 6  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
65bnj1232 28881 . . . . 5  |-  ( ch 
->  n  e.  D
)
76bnj706 28828 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  D )
8 bnj1001.13 . . . . 5  |-  D  =  ( om  \  { (/)
} )
98bnj923 28843 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
107, 9syl 16 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  n  e.  om )
11 elnn 4814 . . 3  |-  ( ( i  e.  n  /\  n  e.  om )  ->  i  e.  om )
124, 10, 11syl2anc 643 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
i  e.  om )
13 bnj1001.5 . . . . . 6  |-  ( ta  <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )
1413simp3bi 974 . . . . 5  |-  ( ta 
->  p  =  suc  n )
1514bnj707 28829 . . . 4  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  p  =  suc  n )
16 nnord 4812 . . . . . . 7  |-  ( n  e.  om  ->  Ord  n )
17 ordsucelsuc 4761 . . . . . . 7  |-  ( Ord  n  ->  ( i  e.  n  <->  suc  i  e.  suc  n ) )
189, 16, 173syl 19 . . . . . 6  |-  ( n  e.  D  ->  (
i  e.  n  <->  suc  i  e. 
suc  n ) )
1918biimpa 471 . . . . 5  |-  ( ( n  e.  D  /\  i  e.  n )  ->  suc  i  e.  suc  n )
20 eleq2 2465 . . . . 5  |-  ( p  =  suc  n  -> 
( suc  i  e.  p 
<->  suc  i  e.  suc  n ) )
2119, 20anim12i 550 . . . 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 1183 . . 3  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e. 
suc  n ) ) )
23 bnj926 28844 . . 3  |-  ( ( suc  i  e.  suc  n  /\  ( suc  i  e.  p  <->  suc  i  e.  suc  n ) )  ->  suc  i  e.  p
)
2422, 23syl 16 . 2  |-  ( ( th  /\  ch  /\  ta  /\  et )  ->  suc  i  e.  p
)
251, 12, 243jca 1134 1  |-  ( ( th  /\  ch  /\  ta  /\  et )  -> 
( ch"  /\  i  e. 
om  /\  suc  i  e.  p ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    \ cdif 3277   (/)c0 3588   {csn 3774   Ord word 4540   suc csuc 4543   omcom 4804    Fn wfn 5408   ` cfv 5413    /\ w-bnj17 28756
This theorem is referenced by:  bnj1020  29040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-bnj17 28757
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