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Theorem sucel 5453
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 2887 . 2  |-  ( suc 
A  e.  B  <->  E. x  e.  B  x  =  suc  A )
2 dfcleq 2417 . . . 4  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  y  e.  suc  A ) )
3 vex 3020 . . . . . . 7  |-  y  e. 
_V
43elsuc 5449 . . . . . 6  |-  ( y  e.  suc  A  <->  ( y  e.  A  \/  y  =  A ) )
54bibi2i 314 . . . . 5  |-  ( ( y  e.  x  <->  y  e.  suc  A )  <->  ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
65albii 1685 . . . 4  |-  ( A. y ( y  e.  x  <->  y  e.  suc  A )  <->  A. y ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
72, 6bitri 252 . . 3  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
87rexbii 2861 . 2  |-  ( E. x  e.  B  x  =  suc  A  <->  E. x  e.  B  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
91, 8bitri 252 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 187    \/ wo 369   A.wal 1435    = wceq 1437    e. wcel 1872   E.wrex 2710   suc csuc 5382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-rex 2715  df-v 3019  df-un 3379  df-sn 3937  df-suc 5386
This theorem is referenced by:  axinf2  8093  zfinf2  8095
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