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Theorem bnj970 29345
Description: Technical lemma for bnj69 29406. 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 29202 . . . 4  |-  ( ch 
->  n  e.  D
)
323ad2ant1 1020 . . 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 1007 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  =  suc  n )
6 bnj970.10 . . . . 5  |-  D  =  ( om  \  { (/)
} )
76bnj923 29166 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
8 peano2 6706 . . . . 5  |-  ( n  e.  om  ->  suc  n  e.  om )
9 eleq1 2476 . . . . 5  |-  ( p  =  suc  n  -> 
( p  e.  om  <->  suc  n  e.  om )
)
10 bianir 970 . . . . 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 5418 . . . . . 6  |-  suc  n  =  ( n  u. 
{ n } )
1413eqeq2i 2422 . . . . 5  |-  ( p  =  suc  n  <->  p  =  ( n  u.  { n } ) )
15 ssun2 3609 . . . . . . 7  |-  { n }  C_  ( n  u. 
{ n } )
16 vex 3064 . . . . . . . 8  |-  n  e. 
_V
1716snnz 4092 . . . . . . 7  |-  { n }  =/=  (/)
18 ssn0 3774 . . . . . . 7  |-  ( ( { n }  C_  ( n  u.  { n } )  /\  {
n }  =/=  (/) )  -> 
( n  u.  {
n } )  =/=  (/) )
1915, 17, 18mp2an 672 . . . . . 6  |-  ( n  u.  { n }
)  =/=  (/)
20 neeq1 2686 . . . . . 6  |-  ( p  =  ( n  u. 
{ n } )  ->  ( p  =/=  (/) 
<->  ( n  u.  {
n } )  =/=  (/) ) )
2119, 20mpbiri 235 . . . . 5  |-  ( p  =  ( n  u. 
{ n } )  ->  p  =/=  (/) )
2214, 21sylbi 197 . . . 4  |-  ( p  =  suc  n  ->  p  =/=  (/) )
2322adantl 466 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  =/=  (/) )
246eleq2i 2482 . . . 4  |-  ( p  e.  D  <->  p  e.  ( om  \  { (/) } ) )
25 eldifsn 4099 . . . 4  |-  ( p  e.  ( om  \  { (/)
} )  <->  ( p  e.  om  /\  p  =/=  (/) ) )
2624, 25bitri 251 . . 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600    \ cdif 3413    u. cun 3414    C_ wss 3416   (/)c0 3740   {csn 3974   suc csuc 5414    Fn wfn 5566   omcom 6685    /\ w-bnj17 29078    FrSe w-bnj15 29084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-tr 4492  df-eprel 4736  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-om 6686  df-bnj17 29079
This theorem is referenced by:  bnj910  29346
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