Theorem onsucssi 4417

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 4343	. 2 |- Ord B
4		ordelsuc 4416	. 2 |- ( ( A e. On /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )
5	1, 3, 4	mp2an 422	1 |- ( A e. B <-> suc A C_ B )

Syntax hints:   <-> wb 104   e. wcel 1480   C_ wss 3066   Ord word 4279   Oncon0 4280   suc csuc 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288

This theorem is referenced by: (None)