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Theorem sucel 4940
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 2979 . 2  |-  ( suc 
A  e.  B  <->  E. x  e.  B  x  =  suc  A )
2 dfcleq 2447 . . . 4  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  y  e.  suc  A ) )
3 vex 3109 . . . . . . 7  |-  y  e. 
_V
43elsuc 4936 . . . . . 6  |-  ( y  e.  suc  A  <->  ( y  e.  A  \/  y  =  A ) )
54bibi2i 311 . . . . 5  |-  ( ( y  e.  x  <->  y  e.  suc  A )  <->  ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
65albii 1645 . . . 4  |-  ( A. y ( y  e.  x  <->  y  e.  suc  A )  <->  A. y ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
72, 6bitri 249 . . 3  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
87rexbii 2956 . 2  |-  ( E. x  e.  B  x  =  suc  A  <->  E. x  e.  B  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
91, 8bitri 249 1  |-  ( suc 
A  e.  B  <->  E. x  e.  B  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 366   A.wal 1396    = wceq 1398    e. wcel 1823   E.wrex 2805   suc csuc 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-v 3108  df-un 3466  df-sn 4017  df-suc 4873
This theorem is referenced by:  axinf2  8048  zfinf2  8050
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