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Theorem sucel 4204
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 2402 . 2  |-  ( suc 
A  e.  B  <->  E. x  e.  B  x  =  suc  A )
2 dfcleq 2079 . . . 4  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  y  e.  suc  A ) )
3 vex 2617 . . . . . . 7  |-  y  e. 
_V
43elsuc 4200 . . . . . 6  |-  ( y  e.  suc  A  <->  ( y  e.  A  \/  y  =  A ) )
54bibi2i 225 . . . . 5  |-  ( ( y  e.  x  <->  y  e.  suc  A )  <->  ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
65albii 1402 . . . 4  |-  ( A. y ( y  e.  x  <->  y  e.  suc  A )  <->  A. y ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
72, 6bitri 182 . . 3  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
87rexbii 2381 . 2  |-  ( E. x  e.  B  x  =  suc  A  <->  E. x  e.  B  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
91, 8bitri 182 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 103    \/ wo 662   A.wal 1285    = wceq 1287    e. wcel 1436   E.wrex 2356   suc csuc 4159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2616  df-un 2990  df-sn 3431  df-suc 4165
This theorem is referenced by: (None)
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