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Theorem bnj556 28677
Description: Technical lemma for bnj852 28698. 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
bnj556.18  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj556.19  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
Assertion
Ref Expression
bnj556  |-  ( et 
->  si )

Proof of Theorem bnj556
StepHypRef Expression
1 vex 2867 . . . . 5  |-  p  e. 
_V
21bnj216 28505 . . . 4  |-  ( m  =  suc  p  ->  p  e.  m )
323anim3i 1139 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  ->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m ) )
43adantr 451 . 2  |-  ( ( ( m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  /\  p  e.  om )  ->  (
m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
5 bnj556.19 . . 3  |-  ( et  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p ) )
6 bnj258 28478 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  p  e.  om  /\  m  =  suc  p )  <->  ( (
m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  /\  p  e.  om ) )
75, 6bitri 240 . 2  |-  ( et  <->  ( ( m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  /\  p  e.  om ) )
8 bnj556.18 . 2  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
94, 7, 83imtr4i 257 1  |-  ( et 
->  si )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   suc csuc 4473   omcom 4735    /\ w-bnj17 28456
This theorem is referenced by:  bnj557  28678  bnj561  28680  bnj562  28681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-un 3233  df-sn 3722  df-suc 4477  df-bnj17 28457
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