Theorem sucel 3032

Description: Membership of a successor in another class.

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

Proof of Theorem sucel

Step	Hyp	Ref	Expression
1		risset 1677	. 2 |- (suc A e. B <-> E.x e. B x = suc A)
2		dfcleq 1463	. . . 4 |- (x = suc A <-> A.y(y e. x <-> y e. suc A))
3		visset 1804	. . . . . . 7 |- y e. V
4	3	elsuc 3028	. . . . . 6 |- (y e. suc A <-> (y e. A \/ y = A))
5	4	bibi2i 606	. . . . 5 |- ((y e. x <-> y e. suc A) <-> (y e. x <-> (y e. A \/ y = A)))
6	5	albii 996	. . . 4 |- (A.y(y e. x <-> y e. suc A) <-> A.y(y e. x <-> (y e. A \/ y = A)))
7	2, 6	bitr 173	. . 3 |- (x = suc A <-> A.y(y e. x <-> (y e. A \/ y = A)))
8	7	rexbii 1660	. 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	bitr 173	1 |- (suc A e. B <-> E.x e. B A.y(y e. x <-> (y e. A \/ y = A)))

Syntax hints:   <-> wb 146   \/ wo 222  A.wal 951   = wceq 953   e. wcel 955  E.wrex 1638  suc csuc 2940

This theorem is referenced by:  axinf2 4596  zfinf 4598

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rex 1642  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-suc 2944

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