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Theorem bnj970 29319
Description: Technical lemma for bnj69 29380. 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
bnj970.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj970.10  |-  D  =  ( om  \  { (/)
} )
Assertion
Ref Expression
bnj970  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  e.  D )

Proof of Theorem bnj970
StepHypRef Expression
1 bnj970.3 . . . . 5  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
21bnj1232 29176 . . . 4  |-  ( ch 
->  n  e.  D
)
323ad2ant1 979 . . 3  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  D )
43adantl 454 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  n  e.  D )
5 simpr3 966 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  =  suc  n )
6 bnj970.10 . . . . 5  |-  D  =  ( om  \  { (/)
} )
76bnj923 29138 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
8 peano2 4866 . . . . 5  |-  ( n  e.  om  ->  suc  n  e.  om )
9 eleq1 2497 . . . . 5  |-  ( p  =  suc  n  -> 
( p  e.  om  <->  suc  n  e.  om )
)
10 bnj926 29139 . . . . 5  |-  ( ( suc  n  e.  om  /\  ( p  e.  om  <->  suc  n  e.  om )
)  ->  p  e.  om )
118, 9, 10syl2an 465 . . . 4  |-  ( ( n  e.  om  /\  p  =  suc  n )  ->  p  e.  om )
127, 11sylan 459 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  e.  om )
13 df-suc 4588 . . . . . 6  |-  suc  n  =  ( n  u. 
{ n } )
1413eqeq2i 2447 . . . . 5  |-  ( p  =  suc  n  <->  p  =  ( n  u.  { n } ) )
15 ssun2 3512 . . . . . . 7  |-  { n }  C_  ( n  u. 
{ n } )
16 vex 2960 . . . . . . . 8  |-  n  e. 
_V
1716snnz 3923 . . . . . . 7  |-  { n }  =/=  (/)
18 ssn0 3661 . . . . . . 7  |-  ( ( { n }  C_  ( n  u.  { n } )  /\  {
n }  =/=  (/) )  -> 
( n  u.  {
n } )  =/=  (/) )
1915, 17, 18mp2an 655 . . . . . 6  |-  ( n  u.  { n }
)  =/=  (/)
20 neeq1 2610 . . . . . 6  |-  ( p  =  ( n  u. 
{ n } )  ->  ( p  =/=  (/) 
<->  ( n  u.  {
n } )  =/=  (/) ) )
2119, 20mpbiri 226 . . . . 5  |-  ( p  =  ( n  u. 
{ n } )  ->  p  =/=  (/) )
2214, 21sylbi 189 . . . 4  |-  ( p  =  suc  n  ->  p  =/=  (/) )
2322adantl 454 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  =/=  (/) )
246eleq2i 2501 . . . 4  |-  ( p  e.  D  <->  p  e.  ( om  \  { (/) } ) )
25 eldifsn 3928 . . . 4  |-  ( p  e.  ( om  \  { (/)
} )  <->  ( p  e.  om  /\  p  =/=  (/) ) )
2624, 25bitri 242 . . 3  |-  ( p  e.  D  <->  ( p  e.  om  /\  p  =/=  (/) ) )
2712, 23, 26sylanbrc 647 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  e.  D
)
284, 5, 27syl2anc 644 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600    \ cdif 3318    u. cun 3319    C_ wss 3321   (/)c0 3629   {csn 3815   suc csuc 4584   omcom 4846    Fn wfn 5450    /\ w-bnj17 29051    FrSe w-bnj15 29057
This theorem is referenced by:  bnj910  29320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-tr 4304  df-eprel 4495  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-bnj17 29052
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