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Theorem bnj563 29341
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 29305 . . . . 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 29296 . . . . 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 252 . . . 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 461 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  ->  n  =  suc  m )
61, 5sylbi 198 . 2  |-  ( et 
->  n  =  suc  m )
7 bnj563.21 . . . 4  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
87simp2bi 1021 . . 3  |-  ( rh 
->  suc  i  e.  n
)
97simp3bi 1022 . . 3  |-  ( rh 
->  m  =/=  suc  i
)
108, 9jca 534 . 2  |-  ( rh 
->  ( suc  i  e.  n  /\  m  =/= 
suc  i ) )
11 necom 2700 . . . 4  |-  ( m  =/=  suc  i  <->  suc  i  =/=  m )
12 eleq2 2502 . . . . . 6  |-  ( n  =  suc  m  -> 
( suc  i  e.  n 
<->  suc  i  e.  suc  m ) )
1312biimpa 486 . . . . 5  |-  ( ( n  =  suc  m  /\  suc  i  e.  n
)  ->  suc  i  e. 
suc  m )
14 elsuci 5508 . . . . . . 7  |-  ( suc  i  e.  suc  m  ->  ( suc  i  e.  m  \/  suc  i  =  m ) )
15 orcom 388 . . . . . . . 8  |-  ( ( suc  i  =  m  \/  suc  i  e.  m )  <->  ( suc  i  e.  m  \/  suc  i  =  m
) )
16 neor 2755 . . . . . . . 8  |-  ( ( suc  i  =  m  \/  suc  i  e.  m )  <->  ( suc  i  =/=  m  ->  suc  i  e.  m )
)
1715, 16bitr3i 254 . . . . . . 7  |-  ( ( suc  i  e.  m  \/  suc  i  =  m )  <->  ( suc  i  =/=  m  ->  suc  i  e.  m ) )
1814, 17sylib 199 . . . . . 6  |-  ( suc  i  e.  suc  m  ->  ( suc  i  =/=  m  ->  suc  i  e.  m ) )
1918imp 430 . . . . 5  |-  ( ( suc  i  e.  suc  m  /\  suc  i  =/=  m )  ->  suc  i  e.  m )
2013, 19stoic3 1656 . . . 4  |-  ( ( n  =  suc  m  /\  suc  i  e.  n  /\  suc  i  =/=  m
)  ->  suc  i  e.  m )
2111, 20syl3an3b 1302 . . 3  |-  ( ( n  =  suc  m  /\  suc  i  e.  n  /\  m  =/=  suc  i
)  ->  suc  i  e.  m )
22213expb 1206 . 2  |-  ( ( n  =  suc  m  /\  ( suc  i  e.  n  /\  m  =/= 
suc  i ) )  ->  suc  i  e.  m )
236, 10, 22syl2an 479 1  |-  ( ( et  /\  rh )  ->  suc  i  e.  m )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   suc csuc 5444   omcom 6706    /\ w-bnj17 29279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-un 3447  df-sn 4003  df-suc 5448  df-bnj17 29280
This theorem is referenced by:  bnj570  29504
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