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Theorem bnj970 33084
Description: Technical lemma for bnj69 33145. 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 32941 . . . 4  |-  ( ch 
->  n  e.  D
)
323ad2ant1 1017 . . 3  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  D )
43adantl 466 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  n  e.  D )
5 simpr3 1004 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  =  suc  n )
6 bnj970.10 . . . . 5  |-  D  =  ( om  \  { (/)
} )
76bnj923 32905 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
8 peano2 6698 . . . . 5  |-  ( n  e.  om  ->  suc  n  e.  om )
9 eleq1 2539 . . . . 5  |-  ( p  =  suc  n  -> 
( p  e.  om  <->  suc  n  e.  om )
)
10 bianir 965 . . . . 5  |-  ( ( suc  n  e.  om  /\  ( p  e.  om  <->  suc  n  e.  om )
)  ->  p  e.  om )
118, 9, 10syl2an 477 . . . 4  |-  ( ( n  e.  om  /\  p  =  suc  n )  ->  p  e.  om )
127, 11sylan 471 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  e.  om )
13 df-suc 4884 . . . . . 6  |-  suc  n  =  ( n  u. 
{ n } )
1413eqeq2i 2485 . . . . 5  |-  ( p  =  suc  n  <->  p  =  ( n  u.  { n } ) )
15 ssun2 3668 . . . . . . 7  |-  { n }  C_  ( n  u. 
{ n } )
16 vex 3116 . . . . . . . 8  |-  n  e. 
_V
1716snnz 4145 . . . . . . 7  |-  { n }  =/=  (/)
18 ssn0 3818 . . . . . . 7  |-  ( ( { n }  C_  ( n  u.  { n } )  /\  {
n }  =/=  (/) )  -> 
( n  u.  {
n } )  =/=  (/) )
1915, 17, 18mp2an 672 . . . . . 6  |-  ( n  u.  { n }
)  =/=  (/)
20 neeq1 2748 . . . . . 6  |-  ( p  =  ( n  u. 
{ n } )  ->  ( p  =/=  (/) 
<->  ( n  u.  {
n } )  =/=  (/) ) )
2119, 20mpbiri 233 . . . . 5  |-  ( p  =  ( n  u. 
{ n } )  ->  p  =/=  (/) )
2214, 21sylbi 195 . . . 4  |-  ( p  =  suc  n  ->  p  =/=  (/) )
2322adantl 466 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  =/=  (/) )
246eleq2i 2545 . . . 4  |-  ( p  e.  D  <->  p  e.  ( om  \  { (/) } ) )
25 eldifsn 4152 . . . 4  |-  ( p  e.  ( om  \  { (/)
} )  <->  ( p  e.  om  /\  p  =/=  (/) ) )
2624, 25bitri 249 . . 3  |-  ( p  e.  D  <->  ( p  e.  om  /\  p  =/=  (/) ) )
2712, 23, 26sylanbrc 664 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  e.  D
)
284, 5, 27syl2anc 661 1  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  e.  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   suc csuc 4880    Fn wfn 5581   omcom 6678    /\ w-bnj17 32818    FrSe w-bnj15 32824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-om 6679  df-bnj17 32819
This theorem is referenced by:  bnj910  33085
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