Theorem ordelsuc 4489

Description: A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)

Assertion

Ref	Expression
ordelsuc	|- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )

Proof of Theorem ordelsuc

Step	Hyp	Ref	Expression
1		ordsucss 4488	. . 3 |- ( Ord B -> ( A e. B -> suc A C_ B ) )
2	1	adantl 275	. 2 |- ( ( A e. C /\ Ord B ) -> ( A e. B -> suc A C_ B ) )
3		sucssel 4409	. . 3 |- ( A e. C -> ( suc A C_ B -> A e. B ) )
4	3	adantr 274	. 2 |- ( ( A e. C /\ Ord B ) -> ( suc A C_ B -> A e. B ) )
5	2, 4	impbid 128	1 |- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   C_ wss 3121   Ord word 4347   suc csuc 4350

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-uni 3797  df-tr 4088  df-iord 4351  df-suc 4356

This theorem is referenced by:  onsucssi  4490  onsucmin  4491  onsucelsucr  4492  onsucsssucr  4493  onsucsssucexmid  4511  frecsuclem  6385  ordgt0ge1  6414  nnsucsssuc  6471  ennnfonelemk  12355  nninfsellemeq  14047