Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj970 Unicode version

Theorem bnj970 29295
Description: Technical lemma for bnj69 29356. 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 29152 . . . 4  |-  ( ch 
->  n  e.  D
)
323ad2ant1 976 . . 3  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  D )
43adantl 452 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  n  e.  D )
5 simpr3 963 . 2  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  p  =  suc  n )
6 bnj970.10 . . . . 5  |-  D  =  ( om  \  { (/)
} )
76bnj923 29114 . . . 4  |-  ( n  e.  D  ->  n  e.  om )
8 peano2 4692 . . . . 5  |-  ( n  e.  om  ->  suc  n  e.  om )
9 eleq1 2356 . . . . 5  |-  ( p  =  suc  n  -> 
( p  e.  om  <->  suc  n  e.  om )
)
10 bnj926 29115 . . . . 5  |-  ( ( suc  n  e.  om  /\  ( p  e.  om  <->  suc  n  e.  om )
)  ->  p  e.  om )
118, 9, 10syl2an 463 . . . 4  |-  ( ( n  e.  om  /\  p  =  suc  n )  ->  p  e.  om )
127, 11sylan 457 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  e.  om )
13 df-suc 4414 . . . . . 6  |-  suc  n  =  ( n  u. 
{ n } )
1413eqeq2i 2306 . . . . 5  |-  ( p  =  suc  n  <->  p  =  ( n  u.  { n } ) )
15 ssun2 3352 . . . . . . 7  |-  { n }  C_  ( n  u. 
{ n } )
16 vex 2804 . . . . . . . 8  |-  n  e. 
_V
1716snnz 3757 . . . . . . 7  |-  { n }  =/=  (/)
18 ssn0 3500 . . . . . . 7  |-  ( ( { n }  C_  ( n  u.  { n } )  /\  {
n }  =/=  (/) )  -> 
( n  u.  {
n } )  =/=  (/) )
1915, 17, 18mp2an 653 . . . . . 6  |-  ( n  u.  { n }
)  =/=  (/)
20 neeq1 2467 . . . . . 6  |-  ( p  =  ( n  u. 
{ n } )  ->  ( p  =/=  (/) 
<->  ( n  u.  {
n } )  =/=  (/) ) )
2119, 20mpbiri 224 . . . . 5  |-  ( p  =  ( n  u. 
{ n } )  ->  p  =/=  (/) )
2214, 21sylbi 187 . . . 4  |-  ( p  =  suc  n  ->  p  =/=  (/) )
2322adantl 452 . . 3  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  =/=  (/) )
246eleq2i 2360 . . . 4  |-  ( p  e.  D  <->  p  e.  ( om  \  { (/) } ) )
25 eldifsn 3762 . . . 4  |-  ( p  e.  ( om  \  { (/)
} )  <->  ( p  e.  om  /\  p  =/=  (/) ) )
2624, 25bitri 240 . . 3  |-  ( p  e.  D  <->  ( p  e.  om  /\  p  =/=  (/) ) )
2712, 23, 26sylanbrc 645 . 2  |-  ( ( n  e.  D  /\  p  =  suc  n )  ->  p  e.  D
)
284, 5, 27syl2anc 642 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   suc csuc 4410   omcom 4672    Fn wfn 5266    /\ w-bnj17 29027    FrSe w-bnj15 29033
This theorem is referenced by:  bnj910  29296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-bnj17 29028
  Copyright terms: Public domain W3C validator