Theorem ordelsuc 4652

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 4651	. . 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 4569	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   C_ wss 3220   Ord word 4507   suc csuc 4510

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-uni 3936  df-tr 4230  df-iord 4511  df-suc 4516

This theorem is used by:  onsucssi  4653  onsucmin  4654  onsucelsucr  4655  onsucsssucr  4656  onsucsssucexmid  4674  frecsuclem  6677  ordgt0ge1  6708  nnsucsssuc  6765  ennnfonelemk  13291  nninfsellemeq  17057