Theorem onsucssi 4506

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 4427	. 2 |- Ord B
4		ordelsuc 4505	. 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 2148   C_ wss 3130   Ord word 4363   Oncon0 4364   suc csuc 4366

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372

This theorem is referenced by: (None)