Theorem onsucssi 4610

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 4529	. 2 |- Ord B
4		ordelsuc 4609	. 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 2202   C_ wss 3201   Ord word 4465   Oncon0 4466   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-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474

This theorem is referenced by: (None)