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Theorem bnj563 32754
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 32719 . . . . 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 32710 . . . . 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 249 . . . 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 460 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  ->  n  =  suc  m )
61, 5sylbi 195 . 2  |-  ( et 
->  n  =  suc  m )
7 bnj563.21 . . . 4  |-  ( rh  <->  ( i  e.  om  /\  suc  i  e.  n  /\  m  =/=  suc  i
) )
87simp2bi 1007 . . 3  |-  ( rh 
->  suc  i  e.  n
)
97simp3bi 1008 . . 3  |-  ( rh 
->  m  =/=  suc  i
)
108, 9jca 532 . 2  |-  ( rh 
->  ( suc  i  e.  n  /\  m  =/= 
suc  i ) )
11 necom 2729 . . . 4  |-  ( m  =/=  suc  i  <->  suc  i  =/=  m )
12 eleq2 2533 . . . . . . 7  |-  ( n  =  suc  m  -> 
( suc  i  e.  n 
<->  suc  i  e.  suc  m ) )
1312biimpa 484 . . . . . 6  |-  ( ( n  =  suc  m  /\  suc  i  e.  n
)  ->  suc  i  e. 
suc  m )
14 elsuci 4937 . . . . . . . 8  |-  ( suc  i  e.  suc  m  ->  ( suc  i  e.  m  \/  suc  i  =  m ) )
15 orcom 387 . . . . . . . . 9  |-  ( ( suc  i  =  m  \/  suc  i  e.  m )  <->  ( suc  i  e.  m  \/  suc  i  =  m
) )
16 neor 2784 . . . . . . . . 9  |-  ( ( suc  i  =  m  \/  suc  i  e.  m )  <->  ( suc  i  =/=  m  ->  suc  i  e.  m )
)
1715, 16bitr3i 251 . . . . . . . 8  |-  ( ( suc  i  e.  m  \/  suc  i  =  m )  <->  ( suc  i  =/=  m  ->  suc  i  e.  m ) )
1814, 17sylib 196 . . . . . . 7  |-  ( suc  i  e.  suc  m  ->  ( suc  i  =/=  m  ->  suc  i  e.  m ) )
1918imp 429 . . . . . 6  |-  ( ( suc  i  e.  suc  m  /\  suc  i  =/=  m )  ->  suc  i  e.  m )
2013, 19sylan 471 . . . . 5  |-  ( ( ( n  =  suc  m  /\  suc  i  e.  n )  /\  suc  i  =/=  m )  ->  suc  i  e.  m
)
21203impa 1186 . . . 4  |-  ( ( n  =  suc  m  /\  suc  i  e.  n  /\  suc  i  =/=  m
)  ->  suc  i  e.  m )
2211, 21syl3an3b 1261 . . 3  |-  ( ( n  =  suc  m  /\  suc  i  e.  n  /\  m  =/=  suc  i
)  ->  suc  i  e.  m )
23223expb 1192 . 2  |-  ( ( n  =  suc  m  /\  ( suc  i  e.  n  /\  m  =/= 
suc  i ) )  ->  suc  i  e.  m )
246, 10, 23syl2an 477 1  |-  ( ( et  /\  rh )  ->  suc  i  e.  m )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   suc csuc 4873   omcom 6671    /\ w-bnj17 32693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-v 3108  df-un 3474  df-sn 4021  df-suc 4877  df-bnj17 32694
This theorem is referenced by:  bnj570  32917
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