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Theorem bnj563 28145
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj563.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
bnj563.21  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
Assertion
Ref Expression
bnj563  |-  ( ( et  /\  rh )  ->  suc  i  e.  m )

Proof of Theorem bnj563
StepHypRef Expression
1 bnj563.19 . . 3  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
2 bnj312 28110 . . . . 5  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  <->  ( n  =  suc  m  /\  m  e.  D  /\  p  e.  om  /\  m  =  suc  p ) )
3 bnj252 28101 . . . . 5  |-  ( ( n  =  suc  m  /\  m  e.  D  /\  p  e.  om  /\  m  =  suc  p
)  <->  ( n  =  suc  m  /\  (
m  e.  D  /\  p  e.  om  /\  m  =  suc  p ) ) )
42, 3bitri 240 . . . 4  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  <->  ( n  =  suc  m  /\  (
m  e.  D  /\  p  e.  om  /\  m  =  suc  p ) ) )
54simplbi 446 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  ->  n  =  suc  m )
61, 5sylbi 187 . 2  |-  ( et 
->  n  =  suc  m )
7 bnj563.21 . . . 4  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
87simp2bi 971 . . 3  |-  ( rh 
->  suc  i  e.  n
)
97simp3bi 972 . . 3  |-  ( rh 
->  m  =/=  suc  i
)
108, 9jca 518 . 2  |-  ( rh 
->  ( suc  i  e.  n  /\  m  =/= 
suc  i ) )
11 necom 2527 . . . 4  |-  ( m  =/=  suc  i  <->  suc  i  =/=  m )
12 eleq2 2344 . . . . . . 7  |-  ( n  =  suc  m  -> 
( suc  i  e.  n 
<->  suc  i  e.  suc  m ) )
1312biimpa 470 . . . . . 6  |-  ( ( n  =  suc  m  /\  suc  i  e.  n
)  ->  suc  i  e. 
suc  m )
14 elsuci 4458 . . . . . . . 8  |-  ( suc  i  e.  suc  m  ->  ( suc  i  e.  m  \/  suc  i  =  m ) )
15 orcom 376 . . . . . . . . 9  |-  ( ( suc  i  =  m  \/  suc  i  e.  m )  <->  ( suc  i  e.  m  \/  suc  i  =  m
) )
16 neor 2530 . . . . . . . . 9  |-  ( ( suc  i  =  m  \/  suc  i  e.  m )  <->  ( suc  i  =/=  m  ->  suc  i  e.  m )
)
1715, 16bitr3i 242 . . . . . . . 8  |-  ( ( suc  i  e.  m  \/  suc  i  =  m )  <->  ( suc  i  =/=  m  ->  suc  i  e.  m ) )
1814, 17sylib 188 . . . . . . 7  |-  ( suc  i  e.  suc  m  ->  ( suc  i  =/=  m  ->  suc  i  e.  m ) )
1918imp 418 . . . . . 6  |-  ( ( suc  i  e.  suc  m  /\  suc  i  =/=  m )  ->  suc  i  e.  m )
2013, 19sylan 457 . . . . 5  |-  ( ( ( n  =  suc  m  /\  suc  i  e.  n )  /\  suc  i  =/=  m )  ->  suc  i  e.  m
)
21203impa 1146 . . . 4  |-  ( ( n  =  suc  m  /\  suc  i  e.  n  /\  suc  i  =/=  m
)  ->  suc  i  e.  m )
2211, 21syl3an3b 1220 . . 3  |-  ( ( n  =  suc  m  /\  suc  i  e.  n  /\  m  =/=  suc  i
)  ->  suc  i  e.  m )
23223expb 1152 . 2  |-  ( ( n  =  suc  m  /\  ( suc  i  e.  n  /\  m  =/= 
suc  i ) )  ->  suc  i  e.  m )
246, 10, 23syl2an 463 1  |-  ( ( et  /\  rh )  ->  suc  i  e.  m )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   suc csuc 4394   omcom 4656    /\ w-bnj17 28084
This theorem is referenced by:  bnj570  28310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-un 3157  df-sn 3646  df-suc 4398  df-bnj17 28085
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