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Theorem bnj970 29760
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
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 29617 . . . 4  |-  ( ch 
->  n  e.  D
)
323ad2ant1 1027 . . 3  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  D )
43adantl 468 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  n  e.  D )
5 simpr3 1014 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  =  suc  n )
6 bnj970.10 . . . . 5  |-  D  =  ( om  \  { (/)
} )
76bnj923 29581 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
8 peano2 6725 . . . . 5  |-  ( n  e.  om  ->  suc  n  e.  om )
9 eleq1 2495 . . . . 5  |-  ( p  =  suc  n  -> 
( p  e.  om  <->  suc  n  e.  om )
)
10 bianir 976 . . . . 5  |-  ( ( suc  n  e.  om  /\  ( p  e.  om  <->  suc  n  e.  om )
)  ->  p  e.  om )
118, 9, 10syl2an 480 . . . 4  |-  ( ( n  e.  om  /\  p  =  suc  n )  ->  p  e.  om )
127, 11sylan 474 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  e.  om )
13 df-suc 5446 . . . . . 6  |-  suc  n  =  ( n  u. 
{ n } )
1413eqeq2i 2441 . . . . 5  |-  ( p  =  suc  n  <->  p  =  ( n  u.  { n } ) )
15 ssun2 3631 . . . . . . 7  |-  { n }  C_  ( n  u. 
{ n } )
16 vex 3085 . . . . . . . 8  |-  n  e. 
_V
1716snnz 4116 . . . . . . 7  |-  { n }  =/=  (/)
18 ssn0 3796 . . . . . . 7  |-  ( ( { n }  C_  ( n  u.  { n } )  /\  {
n }  =/=  (/) )  -> 
( n  u.  {
n } )  =/=  (/) )
1915, 17, 18mp2an 677 . . . . . 6  |-  ( n  u.  { n }
)  =/=  (/)
20 neeq1 2706 . . . . . 6  |-  ( p  =  ( n  u. 
{ n } )  ->  ( p  =/=  (/) 
<->  ( n  u.  {
n } )  =/=  (/) ) )
2119, 20mpbiri 237 . . . . 5  |-  ( p  =  ( n  u. 
{ n } )  ->  p  =/=  (/) )
2214, 21sylbi 199 . . . 4  |-  ( p  =  suc  n  ->  p  =/=  (/) )
2322adantl 468 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  =/=  (/) )
246eleq2i 2501 . . . 4  |-  ( p  e.  D  <->  p  e.  ( om  \  { (/) } ) )
25 eldifsn 4123 . . . 4  |-  ( p  e.  ( om  \  { (/)
} )  <->  ( p  e.  om  /\  p  =/=  (/) ) )
2624, 25bitri 253 . . 3  |-  ( p  e.  D  <->  ( p  e.  om  /\  p  =/=  (/) ) )
2712, 23, 26sylanbrc 669 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  e.  D
)
284, 5, 27syl2anc 666 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619    \ cdif 3434    u. cun 3435    C_ wss 3437   (/)c0 3762   {csn 3997   suc csuc 5442    Fn wfn 5594   omcom 6704    /\ w-bnj17 29493    FrSe w-bnj15 29499
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-pw 3982  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:  bnj910  29761
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