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Theorem sucel 4465
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 2590 . 2  |-  ( suc 
A  e.  B  <->  E. x  e.  B  x  =  suc  A )
2 dfcleq 2277 . . . 4  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  y  e.  suc  A ) )
3 vex 2791 . . . . . . 7  |-  y  e. 
_V
43elsuc 4461 . . . . . 6  |-  ( y  e.  suc  A  <->  ( y  e.  A  \/  y  =  A ) )
54bibi2i 304 . . . . 5  |-  ( ( y  e.  x  <->  y  e.  suc  A )  <->  ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
65albii 1553 . . . 4  |-  ( A. y ( y  e.  x  <->  y  e.  suc  A )  <->  A. y ( y  e.  x  <->  ( y  e.  A  \/  y  =  A ) ) )
72, 6bitri 240 . . 3  |-  ( x  =  suc  A  <->  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
87rexbii 2568 . 2  |-  ( E. x  e.  B  x  =  suc  A  <->  E. x  e.  B  A. y
( y  e.  x  <->  ( y  e.  A  \/  y  =  A )
) )
91, 8bitri 240 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 176    \/ wo 357   A.wal 1527    = wceq 1623    e. wcel 1684   E.wrex 2544   suc csuc 4394
This theorem is referenced by:  axinf2  7341  zfinf2  7343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-un 3157  df-sn 3646  df-suc 4398
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