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Theorem bnj556 33438
Description: Technical lemma for bnj852 33459. 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 3121 . . . . 5  |-  p  e. 
_V
21bnj216 33268 . . . 4  |-  ( m  =  suc  p  ->  p  e.  m )
323anim3i 1184 . . 3  |-  ( ( m  e.  D  /\  n  =  suc  m  /\  m  =  suc  p )  ->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m ) )
43adantr 465 . 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 33241 . . 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 249 . 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 266 1  |-  ( et 
->  si )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   suc csuc 4886   omcom 6695    /\ w-bnj17 33219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-un 3486  df-sn 4034  df-suc 4890  df-bnj17 33220
This theorem is referenced by:  bnj557  33439  bnj561  33441  bnj562  33442
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