Theorem ordelsuc 4601

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 4600	. . 3 |- ( Ord B -> ( A e. B -> suc A C_ B ) )
2	1	adantl 277	. 2 |- ( ( A e. C /\ Ord B ) -> ( A e. B -> suc A C_ B ) )
3		sucssel 4519	. . 3 |- ( A e. C -> ( suc A C_ B -> A e. B ) )
4	3	adantr 276	. 2 |- ( ( A e. C /\ Ord B ) -> ( suc A C_ B -> A e. B ) )
5	2, 4	impbid 129	1 |- ( ( A e. C /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   C_ wss 3198   Ord word 4457   suc csuc 4460

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-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-uni 3892  df-tr 4186  df-iord 4461  df-suc 4466

This theorem is referenced by:  onsucssi  4602  onsucmin  4603  onsucelsucr  4604  onsucsssucr  4605  onsucsssucexmid  4623  frecsuclem  6567  ordgt0ge1  6598  nnsucsssuc  6655  ennnfonelemk  13011  nninfsellemeq  16552