Theorem ordelsuc 4561

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 4560	. . 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 4479	. . 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 2177   C_ wss 3170   Ord word 4417   suc csuc 4420

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-uni 3857  df-tr 4151  df-iord 4421  df-suc 4426

This theorem is referenced by:  onsucssi  4562  onsucmin  4563  onsucelsucr  4564  onsucsssucr  4565  onsucsssucexmid  4583  frecsuclem  6505  ordgt0ge1  6534  nnsucsssuc  6591  ennnfonelemk  12846  nninfsellemeq  16092