Theorem sucel 4513

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

Step	Hyp	Ref	Expression
1		risset 2561	. 2 |- ( suc A e. B <-> E. x e. B x = suc A )
2		dfcleq 2225	. . . 4 |- ( x = suc A <-> A. y ( y e. x <-> y e. suc A ) )
3		vex 2806	. . . . . . 7 |- y e. _V
4	3	elsuc 4509	. . . . . 6 |- ( y e. suc A <-> ( y e. A \/ y = A ) )
5	4	bibi2i 227	. . . . 5 |- ( ( y e. x <-> y e. suc A ) <-> ( y e. x <-> ( y e. A \/ y = A ) ) )
6	5	albii 1519	. . . 4 |- ( A. y ( y e. x <-> y e. suc A ) <-> A. y ( y e. x <-> ( y e. A \/ y = A ) ) )
7	2, 6	bitri 184	. . 3 |- ( x = suc A <-> A. y ( y e. x <-> ( y e. A \/ y = A ) ) )
8	7	rexbii 2540	. 2 |- ( E. x e. B x = suc A <-> E. x e. B A. y ( y e. x <-> ( y e. A \/ y = A ) ) )
9	1, 8	bitri 184	1 |- ( suc A e. B <-> E. x e. B A. y ( y e. x <-> ( y e. A \/ y = A ) ) )

Syntax hints:   <-> wb 105   \/ wo 716   A.wal 1396   = wceq 1398   e. wcel 2202   E.wrex 2512   suc csuc 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-suc 4474

This theorem is referenced by: (None)