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Theorem sucel 4655
Description: Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
Assertion
Ref Expression
sucel  |-  ( suc 
A  e.  B  <->  E. x  e.  B  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
Distinct variable groups:    x, y, A    x, B
Allowed substitution hint:    B( y)

Proof of Theorem sucel
StepHypRef Expression
1 risset 2754 . 2  |-  ( suc 
A  e.  B  <->  E. x  e.  B  x  =  suc  A )
2 dfcleq 2431 . . . 4  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  y  e.  suc  A ) )
3 vex 2960 . . . . . . 7  |-  y  e. 
_V
43elsuc 4651 . . . . . 6  |-  ( y  e.  suc  A  <->  ( y  e.  A  \/  y  =  A ) )
54bibi2i 306 . . . . 5  |-  ( ( y  e.  x  <->  y  e.  suc  A )  <->  ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
65albii 1576 . . . 4  |-  ( A. y ( y  e.  x  <->  y  e.  suc  A )  <->  A. y ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
72, 6bitri 242 . . 3  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
87rexbii 2731 . 2  |-  ( E. x  e.  B  x  =  suc  A  <->  E. x  e.  B  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
91, 8bitri 242 1  |-  ( suc 
A  e.  B  <->  E. x  e.  B  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359   A.wal 1550    = wceq 1653    e. wcel 1726   E.wrex 2707   suc csuc 4584
This theorem is referenced by:  axinf2  7596  zfinf2  7598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-v 2959  df-un 3326  df-sn 3821  df-suc 4588
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