Theorem onsucssi 4598

Description: A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)

Hypotheses

Ref	Expression
onsucssi.1	|- A e. On
onsucssi.2	|- B e. On

Assertion

Ref	Expression
onsucssi	|- ( A e. B <-> suc A C_ B )

Proof of Theorem onsucssi

Step	Hyp	Ref	Expression
1		onsucssi.1	. 2 |- A e. On
2		onsucssi.2	. . 3 |- B e. On
3	2	onordi 4517	. 2 |- Ord B
4		ordelsuc 4597	. 2 |- ( ( A e. On /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )
5	1, 3, 4	mp2an 426	1 |- ( A e. B <-> suc A C_ B )

Syntax hints:   <-> wb 105   e. wcel 2200   C_ wss 3197   Ord word 4453   Oncon0 4454   suc csuc 4456

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459  df-suc 4462

This theorem is referenced by: (None)